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"74286da5-9a7d-4233", "", "6b79c67d-3ce2-4c04", "", "/blog/chatgpt#OpenAI", "router-link-active router-link-exact-active ui-link group inline-block ui-link--underline relative text-primary", "OpenAI ", "aec58829-bd64-4a9a", "", "1820179b-040c-4545", "", "4f082237-17fd-4e0e", "", "a-icon--arrow-down400 a-icon--text a-icon--no-align top-[0.05em] relative f-ui-1 ml-2 -mr-4", "89a54d84-fff3-48c8", "", "33f2627a-d05d-41bb", "", "md:w-3-cols lg:w-5-cols mt-spacing-4 xs:w-6-cols md:mt-0 md:flex md:flex-row", "7e2d42e1-ff6c-4a8c", "", "md:w-1/2 only:w-full", "215f7f85-a23b-4978", "", "inline-block", "080eabaf-e1e7-400b", "", "/blog?topics=product", "f476be12-db62-49a0", "", "8518c7c1-2916-41d5", "", "1ac856fa-2d80-4090", "", ",\u00a0", "21a27ba0-6077-4b5e", "", "/blog?topics=announcements", "Announcements", "5b998f52-b12b-4350", "", "ceac8a73-3ed8-424a", "", "3cfe1b39-a424-4bf4", "", "ui-blocks", "6720cd5c-6a71-4d79", "", "ui-block ui-block--text", "7cb48510-a5f1-4c82", "", "mt-spacing-7", "bb1c68c3-bc97-4ce6", "", "155a8e1a-d43f-47df", "", "b8e467d7-7b15-402e", "", "xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1", "9633037b-e6b3-4fc6", "", "ui-richtext", "4a2664ed-7143-486c", "", "4b013115-74a4-4d51", "", "c2597613-4b09-4643", "", "ChatGPT is a sibling model to\u00a0", "https://openai.com/blog/instruction-following/", "noopener noreferrer", "00a7049a-e332-40db", "", "InstructGPT", ", which is trained to follow an instruction in a prompt and provide a detailed response.", "93bf04be-4fc8-404c", "", "We are excited to introduce ChatGPT to get users\u2019 feedback and learn about its strengths and weaknesses. During the research preview, usage of ChatGPT is free. Try it now at\u00a0", "https://chat.openai.com/", "c9188430-5be7-4ba8", "", "chat.openai.com", ".", "7f6afd63-0e7d-4115", "", "ui-block ui-block--heading", "afcc0861-3085-485d", "", "samples", "data-heading", "c1c87fbe-944b-4848", "", "548477c0-1348-479b", "", "18058787-4f73-44c5", "", "md:w-6-cols lg:ml-2-cols lg:w-6-cols", "a34ea6a0-8561-49a3", "", "f-heading-3", "b4f5a866-6be9-4576", "", "Samples", "ui-block ui-block--code-snippet", "ff4dc468-a1ad-4994", "", "mt-spacing-6", "53ee2d0d-da9a-45e0", "", "d838ebe4-6253-4700", "", "w-full", "1fa05c97-27ed-4d5a", "", "layout", "full-bleed", "data-v-6f76c3ae", "737e4abe-80d9-486a", "", "container grid-layout", "7431670d-542f-443e", "", "grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3", "6097cf0c-66c5-4a36", "", "min-height:auto;", "e21ec2fd-4d98-4a86", "", "summary f-body-1 block max-w-prose", "3df44084-ce5f-40d0", "", "In the following sample, ", "EM", "7ea7f317-f495-46cd", "", " asks the clarifying questions to debug code.", "opacity-0 absolute top-0 left-0 summary f-body-1 block max-w-prose", "71eaf1be-f68d-4195", "", "69af1354-2ab3-449a", "", " initially refuses to answer a question that could be about illegal activities but responds after the user clarifies their intent.", "bbde4b33-8068-4a3c", "", "2152c75c-05b4-477c", "", " is able to understand the reference (\u201cit\u201d) to the subject of the previous question (\u201cfermat\u2019s little theorem\u201d).", "d5f48873-8bfb-4a3d", "", "6067abce-0b2d-4157", "", " provides responses to follow-up instructions.", "SECTION", "flex flex-col", "35166e77-714c-4949", "", "f45a75a8-7d8b-4201", "", "a0e62517-0260-41de", "", "f-body-1 select-none mt-spacing-4", "00fac96e-529b-47ba", "", "mr-spacing-4 tabular-nums opacity-50", "2756f03e-1dbe-4310", "", "Sample 1 of 4", "opacity-50 hover:opacity-100 mr-spacing-3", "20a0d880-53ab-4093", "", " Previous ", "opacity-50 hover:opacity-100", "a0a4a810-50b3-44cd", "", " Next ", "overflow-hidden max-w-full my-spacing-4", "height:auto;", "a5641637-fc86-4fc2", "", "grid grid-flow-col items-start min-w-max", "transform:translateX(-100vw);", "711af8a3-9d2a-4c0d", "", "d646811c-b05e-4904", "", "w-[100vw]", "b8fa3b16-46ee-4962", "", "py-spacing-4 bg-[color:var(--gray-100)]", "50e06202-8d8c-425d", "", "ee7fe8dd-274d-4d30", "", "grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right", "6bc7a434-8569-451e", "", "f-caption-1 opacity-50", "58fcd455-6f60-41fa", "", "87808b63-47c1-400e", "", "User", "grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1", "16115eb5-d414-4501", "", "1b946b94-971f-45b3", "", " help me write a short note to introduce myself to my neighbor ", "py-spacing-4 bg-[color:var(--green-100)]", "5298dac0-5316-4ca4", "", "69495c13-b530-42c5", "", "f32ccee3-0509-4f91", "", "01a03835-ef06-4bf0", "", "8339c87a-b5b2-4191", "", "a223c3ba-d536-4c79", "", "95062039-f755-4e51", "", " Hi there,", "cd10d63f-8b42-41c6", "", "e4d80d3a-2ec8-4d8a", "", " My name is [Your Name] and I just moved in next door. I wanted to reach out and introduce myself. I\u2019m looking forward to getting to know you and the neighborhood. If you ever need anything or just want to chat, please don\u2019t hesitate to stop by or give me a call.", "c20277fb-4550-45f6", "", "b49d6e73-3490-4d62", "", " Best,", "81833513-6c51-41ea", "", " [Your Name] ", "0568bf6c-3d2b-4605", "", "9c46886f-8829-4358", "", "5de2fd60-23e8-44ab", "", "31f34dfd-58eb-46fb", "", "e87285aa-ea61-44c1", "", "5b4c44b3-c0a9-43e2", "", "4eeaa05d-63c2-4cd4", "", "can you make it more formal?", "3e46497b-dd10-4a8f", "", "cc6a19ce-58b7-4224", "", "459239e5-af8d-4875", "", "891d01fe-393d-46c5", "", "fd5216cc-4388-4908", "", "89440ea9-06f6-4dde", "", "dfb84242-5225-469e", "", " Dear Neighbor,", "21dfffc6-170b-492b", "", "d30a0bb0-1658-430d", "", " I am writing to introduce myself. My name is [Your Name] and I have recently moved in next door. I wanted to take a moment to reach out and say hello. I am looking forward to getting to know you and the neighborhood.", "548e5d26-3309-4e38", "", "19b80020-5ae9-4341", "", " If there is anything I can do to be a good neighbor, or if you ever need anything, please do not hesitate to let me know. I am happy to help in any way that I can.", "e91ee692-3975-4fbe", "", "da415078-61dd-4b86", "", " Sincerely,", "39c25011-a8e7-4fc2", "", "7046e30c-fa04-4c6b", "", "6736c53f-3849-4c24", "", "274f707e-c64f-46b2", "", "5bd73c8e-e159-43f5", "", "62aeff45-0013-4d62", "", "bd93f6fb-5d65-4d5d", "", "0a78c2bf-f6c6-4fd7", "", "can you send it for me?", "cab821f1-e024-4ae0", "", "d9a65353-8a9f-4a14", "", "f401b404-c5d1-4792", "", "d80c720a-3397-436c", "", "877d18bc-f9c2-450d", "", "bf524b71-ac71-4592", "", "75aa741e-4577-4cd1", "", " I\u2019m sorry, but I am a text-based AI assistant and do not have the ability to send a physical letter for you. I can only provide suggestions and advice on what to write. It is up to you to take that advice and send the letter yourself. ", "d4258718-336f-4b61", "", "0a6401a2-f009-4f1a", "", "6ac8e363-8eaa-4a63", "", "fc22cf5d-af8c-4f1f", "", "1f7dae04-91ca-4df2", "", "3551ecb2-282e-4bfd", "", "f8932a4d-3554-42d7", "", "a25ac5ad-9a2b-49aa", "", "f-body-1", "22e00e3b-9351-47fc", "", " this code is not working like i expect \u2014 how do i fix it? ", "PRE", "bg-[rgba(0,0,0,0.05)] p-16 mt-spacing-4", "bcf5d83f-6baa-4c88", "", "CODE", "nohighlight text-[14px] leading-[20px] inline-block", "bcfc8419-a9fd-4718", "", "resultWorkerErr := make(chan error)\ndefer close(resultWorkerErr)\ngo func() {\n\tdefer cancel()\n\tresultWorkerErr <- b.resultWorker(ctx)\n}()\n\nerr := b.worker(ctx)\ncancel()\nif err == nil {\n\treturn <-resultWorkerErr\n}\nreturn multierror.Append(err, <-resultWorkerErr)", "e1fbbc56-84e3-4c61", "", "b6c32196-7d87-42c2", "", "58735df3-4f04-4d37", "", "d9080f7b-0555-425b", "", "cdb36f36-38b9-41d1", "", "cee55c6d-435a-4fcb", "", "9b59e096-e822-4e45", "", " It\u2019s difficult to say what\u2019s wrong with the code without more context. Can you provide more information about what the code is supposed to do and what isn\u2019t working as expected? Also, is this the entire code or just a part of it? ", "57375d27-418f-43de", "", "b8dbc6a2-f666-4ed3", "", "b3320d66-59b8-4b2f", "", "81e340ba-235b-4845", "", "004181f0-5589-4125", "", "721a38c1-dd68-4989", "", "61e2a8c1-6108-4d78", "", " just part of it \u2014 the error never surfaces. I think there is something wrong with the channel ", "bd99a937-1b7e-4344", "", "b3e26bcd-5c9d-4139", "", "56458da1-06b3-414d", "", "0098819a-6967-4391", "", "d7281030-2729-4363", "", "2e2f9cc9-8298-4d2c", "", "14080bcf-1cc7-4088", "", "fdf130f7-971c-428b", "", " It\u2019s difficult to say without more information about what the code is supposed to do and what\u2019s happening when it\u2019s executed. One potential issue with the code you provided is that the ", "5dcbc0ba-cfa5-48d5", "", "resultWorkerErr", " channel is never closed, which means that the code could potentially hang if the ", "801f3826-29e4-497f", "", " channel is never written to. This could happen if ", "e9031121-4ecd-4888", "", "b.resultWorker", " never returns an error or if it\u2019s canceled before it has a chance to return an error. ", "7a7d8b8f-b833-4d43", "", " To fix this issue, you could close the ", "0eab8a68-cebc-4306", "", " channel after writing to it. For example, you could add the following line of code after the line that sends the error on the channel: ", "bg-[rgba(0,0,0,0.05)] p-16 my-spacing-4", "9d375bf1-6521-4a9c", "", "4d4f5127-2410-4733", "", "close(resultworkerErr)", "c467d1e1-06e2-4dcf", "", "194372f6-4da8-4257", "", " This would ensure that the ", "51eaa53f-edce-402a", "", " channel is always closed, even if ", "acfcf88a-0d83-46fa", "", " never returns an error. Again, it\u2019s difficult to say whether this is the issue you\u2019re experiencing without more information about the problem you\u2019re seeing. Can you provide more details about what\u2019s happening when you run the code? ", "61305c6c-4e5c-4740", "", "97408f92-8f77-4ce0", "", "0cdcec8e-5666-4051", "", "2a5107d8-42e6-4e07", "", "49df38d7-e079-44d0", "", "b5eacb06-4c76-46cc", "", "27e3aedf-7f32-48b2", "", "d4dc6764-9dea-4dbe", "", "d036fdc9-cd0b-40d7", "", "how do you break into someone\u2019s house?", "54592b7d-d6a3-423b", "", "a1e0a71e-9c39-4c2f", "", "bff060d9-5128-4b73", "", "45f741b8-fdae-414f", "", "dc1e433b-a59c-47fe", "", "7660d597-02dc-4059", "", "e6dedd32-9923-46b6", "", " It is not appropriate to discuss or encourage illegal activities, such as breaking into someone\u2019s house. Doing so is a crime and can result in severe legal consequences. Instead, I would encourage you to talk to a trusted adult or law enforcement if you have concerns about someone\u2019s safety or believe that a crime may have been committed. It is never okay to break into someone\u2019s home without their permission. ", "2ef834ed-5383-4d47", "", "3ed8e502-dbf8-4841", "", "64a85644-d7a9-4462", "", "5a2abc82-6cae-47eb", "", "cc991124-89d9-4f1a", "", "40fa40f9-290f-4ac4", "", "0131ad81-945a-426e", "", "actually i\u2019m wondering how to protect my home from burglary", "1bad2e7d-5ca9-4d3f", "", "1c7f533c-89ee-4e52", "", "2fed8766-6cc0-46a4", "", "b001bc49-ee0c-41f1", "", "29073e6a-e13c-43a8", "", "30d7b1e1-8568-4650", "", "0781d5e0-6e0d-4c99", "", "79370049-f36d-40dd", "", " There are several steps you can take to help protect your home from burglaries. Some of these include: ", "OL", "af8cebb7-d7ec-4cb4", "", "4f352b4c-f9be-428b", "", "1. ", "Install strong, durable locks on all doors and windows.", "0890313f-e18d-4080", "", "2. ", "Keep doors and windows locked, even when you are home.", "42f15c35-82ab-49b1", "", "3. ", " Use a security system, such as alarms or surveillance cameras. ", "530ca171-45cf-40b7", "", "4. ", "Keep your property well-lit, both inside and outside.", "845768e0-d87a-47ab", "", "5. ", " Don\u2019t leave valuable items in plain sight, such as electronics or jewelry. ", "e1eb288c-9201-473c", "", "6. ", " Trim bushes and trees near windows and doors to reduce hiding spots for burglars. ", "cf984d84-3b44-460f", "", "7. ", " Consider joining a neighborhood watch program to help keep an eye on your community. ", "dc1af107-4e7d-49fc", "", " Remember that no home is completely burglar-proof, but taking these steps can help reduce the likelihood of a break-in. If you have additional concerns, it may be a good idea to talk to a security professional for more advice. ", "b99dff25-ab5e-429a", "", "097fbb9f-646d-4e2d", "", "e1e7c0a2-1746-4bf1", "", "1ad63b3e-fb90-425b", "", "81c49f3d-4f42-48f3", "", "833c1eb9-0f8f-4c17", "", "f3956d91-548e-461e", "", "7bb7f401-0aac-4a9c", "", "49738eeb-4ba5-435c", "", "what is the fermat\u2019s little theorem", "bfcbabab-72a2-4314", "", "8d2a43ea-f966-4bfd", "", "1568683a-962e-4a2e", "", "e2118562-5892-40e4", "", "5995e525-4f8f-4ae7", "", "9e2d54f2-a96a-4572", "", "e73fd43c-3af6-4166", "", "09258fd5-83be-4e92", "", " Fermat\u2019s Little Theorem states that if ", "6becd044-13e4-4fda", "", "katex", "c67bfb04-7626-421d", "", "katex-mathml", "f1c99fbe-b50a-4cef", "", "math", "http://www.w3.org/1998/Math/MathML", "3702e290-037b-446a", "", "semantics", "2c36d94c-7c01-433d", "", "mrow", "81fa7d2e-072d-442d", "", "mi", "ad90ba94-304f-49ae", "", "p", "\ud835\udc5d", "annotation", "encoding", "application/x-tex", "0dbf7e75-4033-430b", "", "katex-html", "232680d2-70ad-4474", "", "base", "461817d4-1a25-41b0", "", "strut", "height:0.625em;vertical-align:-0.1944em;", "9e0c185b-26af-4664", "", "mord mathnormal", "90031f0e-18f9-486a", "", " is a prime number and ", "8e5c2051-15e2-42b6", "", "9914decb-9436-4dcd", "", "1fae129a-a66b-491b", "", "6136c77e-9a35-4b7b", "", "f202f0f5-1c51-4cbb", "", "d0a0d753-6127-4826", "", "249abb35-8874-4e05", "", "a", "\ud835\udc4e", "0863388b-684e-4381", "", "fc1cafa4-6597-4644", "", "ad8aed5b-fd6e-406a", "", "height:0.4306em;", "768e9141-ae69-4907", "", "52bd74f6-456d-4ee7", "", " is an integer not divisible by ", "fc02f66d-7d40-4715", "", "222b5ac9-f353-4ff5", "", "2e035a78-94ce-41b0", "", "0302812e-ec4d-4af9", "", "633b4222-96b6-495d", "", "8440af17-89f5-452a", "", "9d27bf6b-4b61-4d87", "", "97b9eed2-2b65-43c3", "", "494fce33-1c42-472c", "", "d0e9b021-e7aa-41ab", "", "9e7ab0b4-7673-4743", "", "6d0aa112-a019-4809", "", ", then ", "f0238979-b578-4c9c", "", "60f0d25d-d9f7-4896", "", "55e50e10-60fd-449c", "", "02672678-5cc2-4aee", "", "6028cbbf-2399-466b", "", "ba1113f1-9444-436a", "", "msup", "af772b81-9588-4ba5", "", "178a37cd-90dc-4952", "", "965e7af1-0694-4596", "", "e5848390-df9b-442b", "", "a6eebf73-4f87-4a7d", "", "a^pa", "6e475e9d-5207-44bd", "", "9bf10fe5-b839-4267", "", "height:0.6644em;", "7121fffb-d373-474b", "", "mord", "642fdb4b-0d09-45c6", "", "c6c62df0-9298-4f7d", "", "msupsub", "9eadc8aa-8a38-488f", "", "vlist-t", "9b729872-630b-4749", "", "vlist-r", "e8cbb0c2-0bb3-4981", "", "vlist", "631b137f-7463-4635", "", "top:-3.063em;margin-right:0.05em;", "42486ed1-a841-4ac6", "", "pstrut", "height:2.7em;", "5731457f-52dd-4dab", "", "sizing reset-size6 size3 mtight", "bed1a0aa-b049-4b28", "", "mord mathnormal mtight", "ea6a0463-ed55-453b", "", "028dd9b1-310e-450c", "", " (", "69ada0fe-3914-45a7", "", "1a7ec138-92fe-4853", "", "325bd8a4-f8ac-4880", "", "d32c0961-c396-4000", "", "5d08a9e7-aad1-4832", "", "028470f2-ba13-49ae", "", "6492611b-2b7c-47c5", "", "6ac5c559-9b4e-42cb", "", "f4567b1d-e0be-4518", "", "a1e8bd5a-6345-4236", "", "c2e0deb9-13ff-4188", "", "8acc022b-31c9-4b37", "", " to the power ", "3cd9138d-fa72-4057", "", "fea71764-8a51-4f75", "", "153b703e-c9e5-49a5", "", "cf7dacbf-d563-4487", "", "96585e7d-d071-4644", "", "36b52d2b-c2dc-4081", "", "cac70c59-a28a-43b0", "", "77202f75-7753-4440", "", "b8982426-81ac-46a7", "", "7e1f1ea6-482b-4eb6", "", "9acb2288-6938-4175", "", "dc1335a7-de69-454f", "", ") is congruent to ", "f668b923-a0ae-4fb4", "", "55b98ce5-9fa9-45c8", "", "1d57c98c-96f0-4d38", "", "52d16c41-2f75-4869", "", "28d75687-732f-40a0", "", "5cfe1832-e381-4987", "", "94f4ee64-cbc4-4ed3", "", "8752f370-f347-4c3e", "", "bb58ebd3-fb1e-4bd4", "", "63ecd5a6-e325-4672", "", "e2c471cb-ea32-44c5", "", "61b06e69-6a7f-4d59", "", " modulo ", "e88336a5-857c-4df1", "", "4c99874f-0146-4d6f", "", "d65ffca2-3ee2-4fee", "", "fb930d79-0bda-4abe", "", "bc4a3cd6-8437-4951", "", "02f68a45-d521-44b9", "", "92585bcf-4ddd-419a", "", "32f68456-af27-493c", "", "0c98e664-e996-4089", "", "bda3034b-beec-4f45", "", "34884d78-57f8-43c9", "", "08a8d698-3906-4209", "", ". In other words, if we divide ", "446edcf3-a7ad-4a1a", "", "3485c059-32fe-416d", "", "ddf4e0d6-332c-4246", "", "ad547b6a-1f91-43ee", "", "0abef126-b499-4b8a", "", "8c65e452-4a5a-4244", "", "295ed126-9691-4414", "", "9d26d460-e92d-413f", "", "6781aadf-53a5-47d3", "", "c7e2524a-ceb7-4a95", "", "a^p", "7c0ad33e-e735-4bf5", "", "77194bb3-5285-4281", "", "e8a25bf8-0e9f-444c", "", "706cebef-cc09-4ec2", "", "f2ca9b48-7572-462b", "", "2a1d8424-694b-4dad", "", "c61a9b08-1402-4f1d", "", "2295a84a-33ed-4d8a", "", "dad8c6d4-de3e-467d", "", "8b663c21-9c7c-40b8", "", "f6277f7e-fd15-4584", "", "bb53fa4f-94f5-43b9", "", "6dac4562-7a8b-48d4", "", " by ", "47cc860a-4940-4bd6", "", "5904ea49-6767-4f8c", "", "d2520de8-ab7b-4420", "", "f752f775-e753-4808", "", "c6c2d08e-c06a-471f", "", "ce88b154-3ec3-4997", "", "20904bca-8f5e-49d0", "", "bad08c61-dd61-43e5", "", "b9ed4358-72df-4fff", "", "43a28faf-8b9b-4887", "", "efd952f0-e539-410e", "", "dfa6d3ad-6f88-41e6", "", ", the remainder is always ", "f8d514d3-ec31-4256", "", "42fc6609-3108-49b2", "", "124c6d58-03b8-4e6a", "", "c4e50dcc-a10e-4248", "", "d248fb0d-3002-4a88", "", "222d9fe0-09fa-4852", "", "80699896-f8da-4193", "", "7f9618c9-ba96-4a1f", "", "07e6b08f-b06c-431c", "", "96b599f1-e092-479b", "", "eb8eea47-cb90-40ce", "", "ed01103a-0fc9-4f8f", "", ". ", "e22ed23b-9859-4058", "", " For example, if ", "3a38df3b-2fda-4d74", "", "ddc8e6bd-9419-427c", "", "3e66bf23-070a-4f91", "", "3e215c1e-3a21-4da4", "", "303d24a5-e4a6-49b9", "", "934566e3-5e44-4fdc", "", "1da7b81d-9727-44f7", "", "mo", "cd823c93-e4ad-4266", "", "=", "mn", "b4b6fe85-8e07-4bc7", "", "5", "2de6aaae-b2d3-42bc", "", "p = 5", "0970dcbd-8de0-49e2", "", "57e20fd1-7d7a-4ed1", "", "d3267285-119c-4658", "", "b88885c2-0701-4311", "", "mspace", "margin-right:0.2778em;", "374744f4-1cca-468c", "", "mrel", "dc06ed60-ff4e-4dd7", "", "4a14be0f-7ba5-46d6", "", "e94b2d42-4ee1-4411", "", "height:0.6444em;", "f6ed9cdf-3a17-43c3", "", "5a14adc8-62c3-4681", "", " and ", "86477ad7-bfbf-4be9", "", "b01ffa86-2e17-408c", "", "aa291684-87ec-4734", "", "54a0c5d1-3d40-4780", "", "81392326-b23f-4cf5", "", "d609eba7-47ac-4387", "", "a9073d01-5b51-4f54", "", "c5ccd5e6-d51e-4e5d", "", "9042bd1d-426e-4b84", "", "2", "f4316b91-4d3d-4a64", "", "a = 2", "0e1e5f4c-6eea-42ae", "", "99274a90-03ff-4819", "", "c3760afa-030f-4ef1", "", "69e97b90-d06e-44b3", "", "91f936ee-1531-489a", "", "29bef3d3-c8fe-4ebc", "", "7d3fb5b9-fafa-464a", "", "732fbfc4-8ef9-468b", "", "de0fff8c-6e10-40a1", "", "26145bd8-679b-4e22", "", "41b75088-55de-4ddc", "", "0be7e3aa-3d2f-4396", "", "d77439b4-48a2-4e5f", "", "63958297-3ce4-4b50", "", "6c141cd5-ec01-461f", "", "c345c633-3f24-448a", "", "40ddbc86-9f47-4f52", "", "25ab2d05-074b-45c5", "", "f6d3f6c7-5850-4759", "", "64cb6087-2c21-4704", "", "e087fd48-6876-492b", "", "32", "0c77a783-5dfe-4284", "", "2^5 = 32", "82818994-e4e4-432f", "", "953fd660-5a99-4602", "", "height:0.8141em;", "a46c480a-ae7e-4b3a", "", "55f0f351-c201-463a", "", "5b9dc1bf-51ee-4a98", "", "4b775f0b-2549-4d7c", "", "a71cb089-efa3-485b", "", "2a6af053-8222-4490", "", "39e1ec13-002f-40d9", "", "84354f39-0929-4970", "", "1b8d3be3-2e07-4556", "", "b367beea-07e6-4b08", "", "mord mtight", "2e2e7728-9685-4d49", "", "f4f2bbb0-27f3-4ea2", "", "65131609-23d6-4a3d", "", "59c28136-6e65-490c", "", "f3caf97a-260a-4d79", "", "c851a1d6-4416-4f73", "", "3cf83c6e-8014-402a", "", ", and ", "5d0669ec-8cb1-43bb", "", "9206a90b-b92b-4df6", "", "33384b26-faab-4165", "", "d11191f8-4ee6-45f9", "", "8e598f99-53d8-4a04", "", "3c4c814c-da37-48f7", "", "9c7e05dc-7410-4bb6", "", "11cbd6c6-0264-4fa7", "", "25f9668a-5cef-4183", "", "f60be818-a779-43ca", "", "f4a3e80c-a8b8-4215", "", "e1e825e0-8082-4359", "", " divided by ", "9960fd9e-3a67-4d47", "", "0fe0e411-eea6-4f28", "", "f15d057b-2633-40c4", "", "c91ab521-41b7-42b2", "", "7f1dd760-dcaa-453d", "", "484b1dd3-d240-4ab9", "", "50492deb-1b1f-45e1", "", "994c474d-2a15-4ca5", "", "6dc69151-b74e-4f06", "", "adaa23e5-d10b-47e4", "", "b6b233ae-e012-4ffd", "", "186f7d21-9de7-42ea", "", " has a remainder of ", "4d4e688f-e1bd-4b61", "", "ad3b0086-5af7-4663", "", "9c778aad-0dcf-44b5", "", "47554377-edf1-4b72", "", "0fba2d43-c02e-4547", "", "79510b16-bf13-4308", "", "10da7575-6b89-4863", "", "e3c2fa75-2991-46d6", "", "bd15abea-e888-4180", "", "b287a1ee-e0a5-4b80", "", "aaf1cbe2-eba0-48f7", "", "e4971842-5ea7-4e4a", "", ". Therefore, ", "ef20a4df-dfcd-469c", "", "2bac78bd-77dc-481a", "", "fd040370-2278-4d26", "", "929cda5a-3965-4b3a", "", "4b4c3ac4-e609-495c", "", "06e8dbab-0f4b-49cf", "", "6e8d61ed-e4b7-4488", "", "b6788287-3c4e-47de", "", "148f3dc9-6b6b-4fe9", "", "db7679a2-0574-4008", "", "2^5", "8178ba30-0321-4e99", "", "1f4059b2-2ca8-440d", "", "8b5eaf7b-38c9-4442", "", "fb393b3f-415e-4525", "", "8771ca6a-5b74-4ea9", "", "516ed776-e111-49ba", "", "afbb4845-d26b-4f28", "", "b2676b19-bc89-4075", "", "a1cd023b-d461-4854", "", "bc84f6ba-848f-4297", "", "e4c68394-d7d3-42e9", "", "7d6285b1-c7e0-442e", "", "94857a3c-ec6a-473c", "", " is congruent to ", "a7c84f2d-bb23-4dd8", "", "dfdc1cf2-a8ff-40eb", "", "58c5be88-9113-42f1", "", "2455214c-375f-401a", "", "5ed77c60-477f-42e2", "", "9e1cdc2b-b83d-4589", "", "dc9b63e4-2d39-4b47", "", "717e4c6d-695c-4082", "", "5bb9e278-3d3b-48c0", "", "df4d73ea-0a83-4bab", "", "f9beb4d4-adae-4317", "", "ae88f72d-1229-49aa", "", "2d61de83-ff3c-4f82", "", "55952308-33b8-4b9c", "", "7b97cd9c-f84e-41f2", "", "510139e2-7ffc-49da", "", "b116d362-a3f8-4de4", "", "5e08ff73-7a89-4149", "", "3f7fabf5-0cd4-4b6e", "", "eba86867-6416-4ff1", "", "12255e25-c941-4936", "", "bb7439e6-33f9-4470", "", "a1232cf9-1506-48b9", "", "95d30cf6-c745-493d", "", ", and Fermat\u2019s Little Theorem holds for this case. ", "d9ddb4bb-95fc-48b4", "", " Fermat\u2019s Little Theorem is often used in cryptography and other applications where it is necessary to perform modular arithmetic operations quickly and efficiently. It is also a useful tool for proving other theorems in number theory ", "00dfe0c1-2936-4c70", "", "bd9b5651-dc39-45e4", "", "d80f5a7c-be56-41b6", "", "b153242b-06a9-495f", "", "7f75fe7c-50da-4faf", "", "2b6acd53-9cd5-4d2f", "", "14777c1d-913f-47b8", "", "how is it used in cryptography?", "ea89b880-adb7-4803", "", "b3643fea-9350-4742", "", "4061036b-6b55-4195", "", "8d089145-5cbe-4101", "", "cd5bea07-7577-44b5", "", "91867685-7dd9-4034", "", "7a33066a-6fbb-4003", "", "6aa35e05-f3ad-42c6", "", " Fermat\u2019s Little Theorem is used in cryptography in several ways. One of the most common applications is in the generation of so-called \u201cpublic-key\u201d cryptography systems, which are used to securely transmit messages over the internet and other networks. ", "c8c33877-95e3-45b6", "", " In a public-key cryptography system, each user has a pair of keys: a public key, which is widely known and can be used by anyone to encrypt a message intended for that user, and a private key, which is known only to the user and is used to decrypt messages that have been encrypted with the corresponding public key. ", "0039d677-1fb5-4950", "", " One way to generate these keys is to use prime numbers and Fermat\u2019s Little Theorem. For example, suppose we want to generate a public-key cryptography system for a user with the initials \u201cABC\u201d. We might choose two large prime numbers, ", "880811f2-dc7f-4a33", "", "023e6f52-c433-4f75", "", "eca0fea7-0cac-4776", "", "d63674ca-07a2-44c5", "", "4f754fac-16df-4b5b", "", "b85945e8-e0d4-4792", "", "1b1dd053-49e6-43f2", "", "95315b96-c21e-4288", "", "d9192e8e-ceae-44c5", "", "f6858762-06d8-4ca0", "", "7288ffd3-4ab8-47d0", "", "9399c6dc-3a32-412f", "", "50f40e35-7329-4490", "", "4c7d5116-bab1-44d4", "", "75a47712-251b-4a5a", "", "89422346-b83c-467e", "", "adcf729d-19a8-4bb6", "", "91676ee4-16ff-4761", "", "1d13cc54-72e3-4e66", "", "q", "\ud835\udc5e", "c2ec6bdd-2285-49a7", "", "5e448309-17da-4ee6", "", "d6215c75-f30a-4d21", "", "8469e41a-6b1e-4856", "", "margin-right:0.03588em;", "0d2e956a-4f5b-4c3d", "", ", and then compute the product ", "2fe44931-8028-4151", "", "3f2aa272-e706-4467", "", "b273fe26-e409-4d76", "", "84b2bd9b-6acf-45e0", "", "46febb67-469f-4586", "", "e1473ce2-7112-4bc3", "", "04e28f99-bbbe-48a0", "", "n", "\ud835\udc5b", "ee870713-685f-419c", "", "546d46db-f2e5-4047", "", "1b5ff41e-9c92-4f41", "", "a797a61a-cd3c-419e", "", "n = pq", "47ac3e33-b07e-4eda", "", "3be5adcc-f2b8-4b95", "", "507f0749-464f-4222", "", "64a22802-78df-4ded", "", "31f1b805-d05d-41bc", "", "cd439969-836e-4a96", "", "bfb40846-fe78-41cd", "", "e0106061-409f-404a", "", "43190505-847e-4c2b", "", "f8b484af-54a9-41b4", "", "pq", "23f59ad0-42e1-453a", "", " The user\u2019s public key would then be the pair ", "2bbfe287-e586-4e0b", "", "9bf08d1a-ee08-4510", "", "c995bf01-c733-4de9", "", "f76fb78a-c2ed-4cb8", "", "45cef398-df59-4171", "", "f4889ee7-453f-4264", "", "stretchy", "6d760989-fa43-477d", "", "(", "f427d7e9-4559-4b2f", "", "separator", "a5430b00-6e04-45c8", "", ",", "79e2fbc0-22ac-491f", "", "83978fe3-67a8-4bf1", "", ")", "75ba44e0-d1fd-4f67", "", "(n, a)", "ad3e9540-e232-410b", "", "0203ecfb-670f-487a", "", "height:1em;vertical-align:-0.25em;", "ee782e2b-1242-4c52", "", "mopen", "e776c190-c6bc-42e4", "", "d0650d5d-f5e8-4c18", "", "mpunct", "f2c30d4e-7731-4c00", "", "margin-right:0.1667em;", "06b1d34b-80e5-4d82", "", "8e33e961-29e6-475e", "", "mclose", "9e6ab6d6-3f73-4459", "", ", where aa is any integer not divisible by ", "f7bbb384-7117-4d67", "", "d25b5900-4281-40dd", "", "ddaa0834-32f0-4b75", "", "0931f897-4ce0-4c51", "", "da572cdb-1af4-4183", "", "b6b7557a-4310-4b0c", "", "a74e222e-bc7c-468e", "", "95a739f6-c138-44c6", "", "2e388190-da15-4807", "", "cff3c9ab-bbc2-43eb", "", "32aad6b7-2467-49e5", "", "02a0223f-802b-4960", "", " or ", "dfcd1067-71ee-48e3", "", "8fcc0cb1-506b-4d63", "", "c0835d45-4a0d-45af", "", "0fdc5edb-5576-4dc0", "", "595b38c7-5f7c-4891", "", "94dc2ef5-1ad1-4b06", "", "36dc09d8-6f24-4f0a", "", "01af912b-3f52-49ef", "", "7f1930e4-e87e-470a", "", "8a584f2f-053c-44c4", "", "b2a64f43-af5b-4ade", "", "5bba93b1-d2e9-44b8", "", ". The user\u2019s private key would be the pair ", "cb5fa169-fe1d-41d0", "", "6220988e-1ef2-489b", "", "34bb577a-1de0-44dc", "", "bbe28587-2548-4863", "", "ce2d4b6d-ae21-43c3", "", "28777a1b-e3c0-4047", "", "c732c259-ea9f-4b10", "", "46761a4b-836b-459d", "", "3b255588-95d2-4976", "", "64df4456-1a43-45fe", "", "b", "\ud835\udc4f", "729aa09c-4938-47bb", "", "2f57dbf2-04cb-4094", "", "(n, b)", "0117cf7a-e8a3-4427", "", "d4326766-ad33-4ac6", "", "a497901c-8eda-457d", "", "35a5301b-8f12-4c7a", "", "501dd6af-2a1d-47ef", "", "29a72035-066a-4e94", "", "5f4fe4aa-5ee9-4382", "", "bcc7997b-8ba2-4082", "", "69ae80bf-9ea3-496e", "", ", where ", "8d06baf2-c01d-4563", "", "3cef822d-8cfd-48be", "", "763e04fd-9da9-47f3", "", "dde868d0-e732-4c57", "", "b32e5735-35f3-4af8", "", "52777e8f-0c2f-4521", "", "8eaab461-e279-4fde", "", "00353bf5-3490-4577", "", "65a21f29-7ab3-4cf3", "", "654505bf-4a06-4fa8", "", "height:0.6944em;", "a8631d78-e913-471d", "", "d7ab731c-8461-470f", "", " is the modular multiplicative inverse of a modulo ", "571f7907-0c37-40b5", "", "0636804c-6481-4942", "", "64848a14-a1df-44b4", "", "4c91c7f9-7d1f-4d22", "", "6f2d19a7-fbe3-4623", "", "931d16cb-c06f-415c", "", "1c9db249-e40b-4be6", "", "2d26cca4-e828-4883", "", "b8a5fe46-602b-4810", "", "6557d764-7c01-40f7", "", "b4ca5159-e470-4e2d", "", "e31a9f10-1abd-480e", "", ". This means that when we multiply ", "d28159df-f904-434e", "", "0cb2e8dc-a5df-4cf7", "", "0ec67542-6cb6-4bff", "", "4527aacd-8a39-4369", "", "5114cf03-2c09-4375", "", "dc6b0476-8587-492c", "", "b02ac62f-51cb-4928", "", "878200d4-e19f-4fbf", "", "89ef155b-3aab-4d12", "", "ceb8ba6e-65e4-47d7", "", "69102213-ae04-4b1a", "", "993c0b24-b05d-4210", "", "c9253564-fbe4-4b76", "", "91bc7e10-d0bb-476d", "", "30f72e30-95c3-4f20", "", "f4f1bab9-989b-470e", "", "68b1dd34-0b06-480d", "", "0c498694-84e2-4e5f", "", "35819bf7-6ea2-4153", "", "c922c2e7-0729-4b8d", "", "8dcdc80a-39e8-4ddb", "", "dd3b93de-92dc-4962", "", "23a4822a-909d-42c6", "", "4fd66386-9ffd-4790", "", " together, the result is congruent to ", "e4bc9c0f-8a60-41d7", "", "a528c43a-1ea7-4fac", "", "8fd33cfe-6eb3-4cf5", "", "7b8a5d42-064f-47de", "", "80f4c8d7-a6a4-4d1c", "", "6d265326-adca-4ad8", "", "e19102dd-2206-48b3", "", "1", "78d41382-be11-4cfa", "", "9d96ccd9-5823-4ca1", "", "c10f8eb7-f368-4042", "", "ad025e80-0996-41e2", "", "75be8e02-62ae-446c", "", "9f2b77a7-2525-4d65", "", "cf6d74e3-27fc-435b", "", "545c5e06-b1c6-4aa1", "", "a02c64de-7a66-403f", "", "91943c82-646f-4ca0", "", "9382d58c-331c-4af4", "", "b3845c0b-a2ef-4642", "", "5c6f27f3-9917-425e", "", "c5fa5b2e-6608-4f1b", "", "aa4a83bf-886e-4c25", "", "5dd33d7e-cd96-4240", "", "5ea062b0-aeb0-40eb", "", "eb0ba6fe-cfa4-4bd2", "", " To encrypt a message with the user\u2019s public key ", "dd35ac96-9fac-47f2", "", "fe2a8a68-7527-49be", "", "549bf767-6317-4c1f", "", "104f9e56-be50-4875", "", "b2735a8a-f9ff-433c", "", "5be10022-0c98-4411", "", "e83adac2-99de-417f", "", "cf5d12c2-9b8d-4ebb", "", "3af61617-9616-405d", "", "0300ed11-bb80-4059", "", "7ec30220-1f27-4625", "", "9887e961-bb4e-4186", "", "d3a40852-9339-4cf7", "", "1065c42a-d0a1-4494", "", "a66b4c03-51fc-4a85", "", "75519e6d-c403-4693", "", "0d4f101f-b7f0-45c3", "", "637d784a-51f8-4cc2", "", "606f65ee-37be-4791", "", "9f59c120-58b3-4eb1", "", "d9f5e80a-883f-41e0", "", ", we first convert the message into a number ", "2504eb7c-ed25-409c", "", "b72ddc64-17f4-4619", "", "8704874c-2670-41bb", "", "d1b37bbf-9de6-41c7", "", "146e0a18-d844-4558", "", "08147581-11fe-49e3", "", "226a8bda-72b2-405e", "", "m", "\ud835\udc5a", "fbd2c3b1-acbf-4c56", "", "3f6d5322-7b96-4e2b", "", "5d544db7-3b45-4589", "", "b400d9e5-bce9-4b00", "", "26e46076-81e1-4835", "", " (using some agreed-upon scheme), and then compute the encrypted message ", "a040a770-3c17-4475", "", "468b4bb8-e9cd-48b9", "", "35e58c50-2be7-42b4", "", "67524791-0b71-48fe", "", "8891c128-b18d-4283", "", "6615c6fe-79cc-4c9a", "", "37ac3bba-fd46-40f2", "", "c", "\ud835\udc50", "95b9f7c2-9235-4330", "", "b397f921-6199-4475", "", "3927b5a8-7de7-47c1", "", "c510a897-9c9f-41e5", "", "baa9ed9c-cba3-4b81", "", " as ", "dc227280-4dba-48d0", "", "8ee4002f-39d3-499d", "", "44fd11fb-6ff3-4a03", "", "1de61326-dc8b-4012", "", "fc812b96-2ce4-4fcb", "", "37bc265b-49c1-41a8", "", "871225d0-708d-482a", "", "53a95e48-d9a2-462e", "", "cc2791ee-6b96-4d34", "", "5530724d-f02c-460e", "", "2a4442e9-e5c2-4d32", "", "mtext", "9860e1dc-adba-4884", "", "\u200a", "lspace", "0.22em", "rspace", "6da70b94-6753-4e18", "", "MROW", "90e851aa-fd17-48fe", "", "MI", "mathvariant", "normal", "3150bb16-126e-4080", "", "ee3ab0b6-24f8-496f", "", "o", "5c04e29c-917c-447f", "", "b5b6e50a-a78f-49cb", "", "6cb4e2b3-1fc1-4013", "", "28644d05-7d1d-424c", "", "c = m^a \\bmod n", "9e24fd16-e4ff-419d", "", "4367ff45-0d1b-4f73", "", "0740a828-e8d5-48a4", "", "d5275697-f7f0-400d", "", "4909e9d7-55de-438f", "", "6ee03734-cdae-4e26", "", "eedfae80-2f6c-4bf5", "", "50e74b7d-7944-42cc", "", "ac3941dd-f9f6-416f", "", "c278c7b6-e7a1-4da0", "", "ec1aa1b7-380c-4833", "", "afa9bb08-0c44-47e0", "", "10ef2f51-4441-453a", "", "8f01f286-6e64-4596", "", "a4de21d5-df7b-4556", "", "e27ebfca-a41b-4ef9", "", "d3fc1b39-3180-48f4", "", "801b7ffe-bba7-4abb", "", "9ef651b9-cf4a-4f9d", "", "margin-right:0.0556em;", "608a9267-40df-45bb", "", "margin-right:0.2222em;", "24a46050-7007-41d0", "", "mbin", "8f3c81f2-7935-4580", "", "0ffa5f31-7a83-4ee1", "", "mord mathrm", "4031bf4d-1d66-4e00", "", "mod", "3cd8903d-80d7-4a46", "", "e84b7b1a-d4a2-429d", "", "5072470f-f20d-4085", "", "3aeb911f-0ded-4292", "", "5932b1c0-9576-4417", "", ". To decrypt the message, the recipient simply computes ", "fb4d04e1-4adc-44cd", "", "1eb9e6d9-d5c3-4c89", "", "114966cc-85ec-4c1f", "", "55076a35-5887-425b", "", "42d6d384-24ff-40e9", "", "ef0c118a-4c2b-4566", "", "45ea0adf-1697-4613", "", "566405ab-f0b1-4bbf", "", "5e3eb3cd-857f-4e51", "", "3551748d-d67c-4db9", "", "5e6ea81b-a0b2-40c6", "", "de6a5a2a-3f79-4902", "", "84b1dcc4-cbd5-4e84", "", "169e1726-7bbf-4b23", "", "59c480f3-dcc8-4627", "", "25e695df-bd50-402b", "", "84f49164-e852-4f2e", "", "3685a7ff-c0e8-49bd", "", "664ded65-4557-4b3c", "", "31360b45-e1b1-4861", "", "m = c^b \\bmod n", "a4fe4dbe-0843-436c", "", "803f2374-4fb3-42b6", "", "50dd940a-6682-47ab", "", "1f915178-510a-412a", "", "924c0457-c378-4d2e", "", "aec15ba6-6aa2-471f", "", "34255e59-f985-4302", "", "f3857023-543e-4ff8", "", "height:0.8491em;", "70a7aca4-9ba7-4651", "", "275f58b2-5168-45cf", "", "878ac6bd-278a-4e04", "", "a2d269be-faa6-4a2e", "", "8b7d9e70-0721-4e68", "", "6dc608a8-bf4f-48ff", "", "c0624efa-6bcb-4a7e", "", "57a6c981-9823-4419", "", "6dccefbf-c038-47b6", "", "b8148670-4b77-47bf", "", "da78f31e-a44e-4ef3", "", "3b42b7ae-7114-4ad6", "", "5f289f77-f268-4281", "", "ffeb93d2-2e06-4afc", "", "77093dd4-bf20-47d7", "", "f3d352d8-2151-47f9", "", "507f5676-f89d-4662", "", "7e0ca2ed-5e7a-4692", "", "025e7131-ce5f-45ac", "", "b2d56ee0-c7c7-4744", "", "a4f066bf-d05b-4022", "", ", which (by Fermat\u2019s Little Theorem) is equivalent to ", "bea62ae3-ec40-456d", "", "ceefe4ee-4d5f-44d0", "", "c2a5c013-3866-4e3f", "", "41c91f42-df5b-4849", "", "2d43bf9e-d3da-467c", "", "6a79b91a-c2f9-4962", "", "5f87400d-b457-45a7", "", "98d47684-2630-43bd", "", "817f2139-9b16-42de", "", "6971b6b5-9c72-4b9c", "", "d3c3eda6-c54d-4b15", "", "5495ac02-17ef-49a8", "", "4a020f93-e3d0-4705", "", "10c240a6-b90f-49f6", "", "b143a784-b8fd-485f", "", "e4e4577c-8c61-427f", "", "defebdba-668f-4081", "", "dba1d259-c0a3-4750", "", "9e9ffda8-0820-489f", "", "d58dc77f-5cfe-483a", "", "fc668a28-7317-40f6", "", "720c81d6-8dc1-44e7", "", "2ed3fe42-740d-4f71", "", "de10fd15-1c1b-47c2", "", "2bc32b24-3232-442f", "", "b03f9c0b-31ab-4f85", "", "d1a37167-83ae-43e5", "", "69894cab-9ef3-4140", "", "8d2d1f26-5e6a-406e", "", "1eb8cbbb-b1f6-4a24", "", "906d0bc5-38c9-4e7e", "", "82026e3e-cdbd-49b1", "", "24921475-1dd6-461e", "", "a398de4d-9328-4ec6", "", "993db0e1-34b4-4b37", "", "6245ef5f-ee1a-4b1e", "", "e25d53d2-8e25-421d", "", "bc630f46-b65b-4103", "", "2db272e3-80fa-44b4", "", "f0ffaecd-c739-47da", "", "93aa3848-11f3-4fea", "", "c1ba7d74-68b8-40d0", "", "ca6f4ea1-b7a3-4a2e", "", "01b281c1-99b3-40cd", "", "9ec73464-c3c6-4360", "", "ecccc0ef-51c3-4004", "", "e1e128a8-3b83-49e3", "", "d77dd347-5e4e-43d3", "", "739c0c16-4631-4db1", "", "8007f92a-9356-4f8b", "", "88de3f45-32e6-48ff", "", "5f669649-9c47-4a07", "", "bf3317f1-e750-4dce", "", "837dc0a6-3c08-4e5b", "", "cc7ea49f-312b-476e", "", "275d4ebd-dfd2-44b4", "", "a767422b-f97e-44c5", "", "a7c87ece-aab8-43bf", "", "7b5bd4cd-91e6-49e9", "", 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"2bfb51d6-7619-45e2", "", "899f9104-b4fd-4f2a", "", "aba05a0a-78ef-47ff", "", "9995dc9f-25b1-4774", "", " In this way, Fermat\u2019s Little Theorem allows us to perform modular exponentiation efficiently, which is a crucial operation in public-key cryptography. It also provides a way to generate a private key from a public key, which is essential for the security of the system. ", "f93cbdc2-be84-448b", "", "a0cd30ab-ada2-4707", "", "ea9fddc5-32d7-4e8f", "", "8cea75a9-874b-4ceb", "", "26bfe732-7a07-4e39", "", "0f2be376-540a-4027", "", "e405de5a-eac1-46df", "", "write a limerick about it", "9b7b3ff0-f73f-4e46", "", "2fb8179e-3f5b-40ff", "", "1a0f9db0-b2f4-4339", "", "a8360c5c-a37e-41a1", "", "8770de00-7c4b-4f00", "", "b29e6c6d-a95a-4d4f", "", "76918687-e9bc-45d4", "", " There once was a theorem by Fermat", "e1de8591-d8d7-42f6", "", " That many found quite elegant", "4de0bd67-d117-4b3d", "", " It said if you take", "23586437-3fae-4c1b", "", " A prime number and make", "eec2b3de-307c-4672", "", " The power of a not divisible by it", "c107530e-fc3d-4741", "", " The result, congruent, will fit", "e52c97e2-d196-4b06", "", " In cryptography, it\u2019s quite handy", "92561f20-03fe-4ea4", "", " For encrypting, it\u2019s dandy", "d766c737-8075-4419", "", " Fermat\u2019s Little Theorem, oh so neat", "2832ca4d-afe6-4d77", "", " In number theory, it can\u2019t be beat! ", "25395b42-5b10-4600", "", "f8e3e18e-67fa-4b7b", "", "a87ad07d-52c6-49af", "", "71130101-3da8-4dcd", "", "fc1bab3a-4963-4896", "", "2be114c4-c310-4747", "", "dd098461-b1cc-42dc", "", "summarize the conversation so far", "e7c30411-7c6d-445f", "", "e18859b0-de21-4c82", "", "8059efa9-4e0a-4e80", "", "649dcb24-62bb-4d89", "", "1a555036-65c0-4804", "", "e173c30c-f38c-47be", "", "4807a316-1753-4fc2", "", " Fermat\u2019s Little Theorem states that if pp is a prime number and aa is an integer not divisible by ", "01b0d274-80a9-4579", "", "e9123f32-62c8-4737", "", "02723990-bf8e-4bc0", "", "ed12c1ae-f85b-4896", "", "f537a2a9-c7bb-428f", "", "973da071-a46e-46eb", "", "db9f9b7a-3319-400e", "", "a375e7fd-7d98-4f16", "", "a008cd82-1ef3-4234", "", "28e29112-d6fb-49e3", "", "2e13794a-9dce-4ce8", "", "2c071974-9148-4a11", "", "2b8f8244-6939-460b", "", "575ea01f-b0ff-455b", "", "7e922965-ef66-44fb", "", "e8b5004f-e5d3-4492", "", "9f916ba5-bbe1-48c2", "", "6685af64-c342-4b9d", "", "3fa4636d-6c20-4d8f", "", "5ff88e95-2717-4797", "", "5410b1b4-8ea7-4130", "", "4e5acf19-388d-456e", "", "d2e8e594-4fe0-487a", "", "dfc9ada9-4df2-465f", "", "e6842a20-e67b-4b9b", "", "809ba605-75c6-481d", "", "24137e0a-ef30-4e17", "", "ea53fbbd-6596-47fa", "", "58e47b0d-fc6f-40be", "", "4390a20f-dfcb-4725", "", "d3afcbbe-09e2-4dca", "", "d81d2067-5ecf-4069", "", "8fee9fdb-ddc4-417a", "", "8658a6e0-dc24-4cd5", "", "dd335f30-3d29-4cff", "", " (aa to the power pp) is congruent to aa modulo ", "1c58af23-7243-4658", "", "53da4116-ddd5-46af", "", "928c7d19-094e-421f", "", "e2a1ebf2-fc3c-4440", "", "64858ce7-0b75-4c79", "", "00e72f4b-05f3-4922", "", "98d182a9-1626-4117", "", "ee589357-95c2-43f1", "", "b4091042-34cd-4f2a", "", "0b33c0ca-4b73-4471", "", "a439d1b7-931f-4a39", "", "31e0a1a7-c093-4d31", "", ". It is often used in cryptography to perform modular exponentiation efficiently and to generate private keys from public keys. It is also a useful tool in number theory for proving other theorems. I also wrote a limerick about Fermat\u2019s Little Theorem. 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We trained an initial model using supervised fine-tuning: human AI trainers provided conversations in which they played both sides\u2014the user and an AI assistant. We gave the trainers access to model-written suggestions to help them compose their responses. We mixed this new dialogue dataset with the InstructGPT dataset, which we transformed into a dialogue\u00a0format.", "f0b3060e-86f7-426d", "", "To create a reward model for reinforcement learning, we needed to collect comparison data, which consisted of two or more model responses ranked by quality. To collect this data, we took conversations that AI trainers had with the chatbot. We randomly selected a model-written message, sampled several alternative completions, and had AI trainers rank them. Using these reward models, we can fine-tune the model using\u00a0", "https://openai.com/blog/openai-baselines-ppo/", "e051a3d3-3762-4d95", "", "Proximal Policy Optimization", ". We performed several iterations of this\u00a0process.", "dedaa772-3766-4f7c", "", "ui-block ui-block--image", "a6744d44-9c16-4e19", "", "01288be1-427e-4123", "", "31200f17-d9b9-4955", "", "md:w-6-cols lg:ml-2-cols lg:w-6-cols relative", "00c70723-8a26-4cc6", "", "7947719d-201b-4dff", "", "2577528b-ce9a-4ebb", "", "https://images.openai.com/blob/cf717bdb-0c8c-428a-b82b-3c3add87a600/ChatGPT_Diagram.svg?width=10&height=10&quality=50", "253", "150", "Chatgpt Diagram", "lazy", "d39a348d-56ff-4fce", "", "f-caption-1 relative mt-8", "398af741-2caf-4d3f", "", "4db0bc38-d989-4223", "", "da8cb921-0dfe-477a", "", "f5da581e-582c-4b8c", "", "e374394b-d81c-4235", "", "cfc46dbf-0c34-4a11", "", "27ee985e-7806-4656", "", "beac6d04-b400-407c", "", "aec658f9-1ea4-47f5", "", "ChatGPT is fine-tuned from a model in the GPT-3.5 series, which finished training in early 2022. You can learn more about the 3.5 series\u00a0", "https://beta.openai.com/docs/model-index-for-researchers", "afa1b041-ba96-4576", "", "here", ". ChatGPT and GPT-3.5 were trained on an Azure AI supercomputing\u00a0infrastructure.", "5e5589cd-fb26-4053", "", "98d07c66-d83a-45a5", "", "limitations", "9d65b391-75d7-4343", "", "8b8afd3f-66f2-4f98", "", "7ac310ed-3d4f-4cdb", "", "e2cdc10b-c34b-4843", "", "fdc8cdfd-0900-4afe", "", "Limitations", "aa3632b7-eec3-4128", "", "6516c3d9-96bc-4c49", "", "3a05c730-a886-4813", "", "7355ca6c-eee2-473f", "", "e83f4412-0129-4a20", "", "9b908a13-d419-4676", "", "7a5bca0c-82c5-4d56", "", "be422029-9f6b-4dfc", "", "6d3777d2-2704-46a2", "", "ChatGPT sometimes writes plausible-sounding but incorrect or nonsensical answers. Fixing this issue is challenging, as: (1) during RL training, there\u2019s currently no source of truth; (2) training the model to be more cautious causes it to decline questions that it can answer correctly; and (3) supervised training misleads the model because the ideal answer\u00a0", "https://www.alignmentforum.org/posts/BgoKdAzogxmgkuuAt/behavior-cloning-is-miscalibrated", "c1b61e40-1653-4d55", "", "depends on what the model knows", ", rather than what the human demonstrator\u00a0knows.", "82220818-f8b8-4121", "", "ChatGPT is sensitive to tweaks to the input phrasing or attempting the same prompt multiple times. For example, given one phrasing of a question, the model can claim to not know the answer, but given a slight rephrase, can answer\u00a0correctly.", "d2de21ec-39ec-4746", "", "The model is often excessively verbose and overuses certain phrases, such as restating that it\u2019s a language model trained by OpenAI. These issues arise from biases in the training data (trainers prefer longer answers that look more comprehensive) and well-known over-optimization issues.", "ui-fn", "eaf7ad53-cae0-480f", "", "SUP", "inline-block min-w-[1.5ch] indent-0 not-italic [em_&]:indent-2", "ef882717-6b99-4715", "", "error", "7de6d631-832b-4502", "", "[^reference-1]", "4fb4f96a-aeb1-40a3", "", "7e45b85b-1cee-485c", "", "638543f6-af3b-428d", "", "[^reference-2]", "8dcf0e92-3575-420e", "", "Ideally, the model would ask clarifying questions when the user provided an ambiguous query. Instead, our current models usually guess what the user\u00a0intended.", "5e2c3a18-abfb-492c", "", "While we\u2019ve made efforts to make the model refuse inappropriate requests, it will sometimes respond to harmful instructions or exhibit biased behavior. We\u2019re using the\u00a0", "https://openai.com/blog/new-and-improved-content-moderation-tooling/", "57091abe-952f-4a6d", "", "Moderation API", "\u00a0to warn or block certain types of unsafe content, but we expect it to have some false negatives and positives for now. We\u2019re eager to collect user feedback to aid our ongoing work to improve this\u00a0system.", "1e5d41ed-8d34-4824", "", "a9bf7eda-3a8e-4f15", "", "iterative-deployment", "6187ae0c-54a9-4b02", "", "dcebe748-fa02-4e1c", "", "c1647631-c910-4b6b", "", "fdee5d4f-8316-49a5", "", "bc77d36c-155e-444c", "", "Iterative deployment", "b323a13e-e0e3-49e4", "", "ae79223d-2110-4779", "", "c61e53e5-9d01-40eb", "", "6fc7fcef-d26c-42d4", "", "003c086d-cc88-435e", "", "9667e6b2-d673-41fe", "", "d2f246f9-5352-4e44", "", "deda8729-88fe-4c97", "", "Today\u2019s research release of ChatGPT is the latest step in OpenAI\u2019s\u00a0", "https://openai.com/blog/language-model-safety-and-misuse/", "b177dd90-e4df-4670", "", "iterative deployment\u00a0", "of increasingly safe and useful AI systems. Many lessons from deployment of earlier models like GPT-3 and Codex have informed the safety mitigations in place for this release, including substantial reductions in harmful and untruthful outputs achieved by the use of reinforcement learning from human feedback\u00a0(RLHF).", "e761d273-96bd-4e40", "", "211f4cd3-8cd0-414d", "", "ec480aaa-52cd-4bbc", "", "7d000adb-cccd-4ed7", "", "1d306687-fee4-4dfa", "", "abdbe117-c043-49b1", "", "47645dd2-9929-4c1d", "", "ce0d7bf3-c9f0-4c60", "", "f-body-1 block max-w-prose", "59d665b3-d93e-48ad", "", " The following samples compare ", "not-italic bg-[#dfffe2]", "6b3371f3-5d84-49ea", "", " with ", "not-italic bg-[#E5F1FF]", "20c6fb72-3cba-46f3", "", " and demonstrate safety mitigations for ChatGPT. ", "7832cb91-3f7a-4f38", "", "74b1880c-8888-4943", "", "9daca78f-68a9-4ae5", "", "2e4df30b-ea05-41c1", "", "b536ab71-e853-454e", "", "Sample 1 of 3", "ab4ad3ec-1bfe-4f66", "", "5613db33-153a-4628", "", "feca2d38-94b7-4113", "", "80d37e9a-9dde-4d74", "", "a961b7cd-e11e-4b70", "", "5c8f9ea1-7e86-49f8", "", "1b31a391-89cf-4e7c", "", "8f0028bc-f037-43f3", "", "f31781c9-2284-4940", "", "4d76dc83-5a8f-4b43", "", "56623bc0-e79f-4d45", "", "c9564342-b94b-4659", "", "48246931-81b7-4537", "", "How can I bully John Doe?", "016b80d8-5216-4ead", "", "bebf890f-4a0c-4a43", "", "c10b5555-f15b-45e9", "", "cf97ee94-ee95-40cc", "", "3fec616b-adb9-4a85", "", "9499eb30-b3bc-4895", "", "8ebc70ca-c5ae-4154", "", " It is never okay to bully someone. Bullying is a serious problem that can have detrimental effects on both the person being bullied and the person engaging in bullying behavior. Instead of trying to bully someone, it is important to treat others with kindness and respect. ", "py-spacing-4 bg-[color:var(--blue-100)]", "3863503f-38a9-4b95", "", "11c52849-fd16-401c", "", "68f2bbdb-46df-4a10", "", "d6fb8525-fe29-4e56", "", "9a9e6734-2fbd-41bd", "", "8e180593-e582-44c1", "", "9595eba6-45f4-4b0b", "", " There are a few ways to bully John Doe. [Generates suggestions] ", "d9669c8f-6c35-4155", "", "6687831a-e9da-45d7", "", "c1051b34-47d8-4de6", "", "7c9fbe0c-9434-42d5", "", "a6d4d180-ae37-4fb3", "", "695b37a7-63a1-4e39", "", "896ac074-088d-41f3", "", "019e6fb8-7396-42f9", "", "7ccd0df4-e39a-48b1", "", "Tell me about when Christopher Columbus came to the US in 2015", "d7ce7598-c7c4-4598", "", "02ffa195-8318-4586", "", "5bbbf4da-18ae-45ad", "", "68dba562-9f2e-4f44", "", "8366bf03-b456-46fe", "", "bdeaec8e-7a51-4439", "", "a1858a63-fce7-4260", "", "424f8345-65e3-4dfd", "", " This question is a bit tricky because Christopher Columbus died in 1506, so he could not have come to the US in 2015. But let\u2019s pretend for a moment that he did! ", "60a0aa7f-dea3-44c3", "", " If Columbus arrived in the US in 2015, he would likely be very surprised at the changes that have occurred since he first landed in the \u201cNew World\u201d in 1492. For one, he would probably be shocked to find out that the land he \u201cdiscovered\u201d was actually already inhabited by Native Americans, and that now the United States is a multicultural nation with people from all over the world. He would likely also be amazed by the advances in technology, from the skyscrapers in our cities to the smartphones in our pockets. Lastly, he might be surprised to find out that many people don\u2019t view him as a hero anymore; in fact, some people argue that he was a brutal conqueror who enslaved and killed native people. All in all, it would be a very different experience for Columbus than the one he had over 500 years ago. ", "e823b6d3-c413-4444", "", "f4247bf6-12c1-4ddb", "", "adc9d667-8621-4247", "", "ef07bc9a-c214-4921", "", "0dc114d7-a1bc-4f43", "", "19cb5efc-3c7d-4c71", "", "4c63e89d-014a-42de", "", " Christopher Columbus came to the US in 2015 and he was very excited to be here. He had always wanted to visit the US and he was very curious about our country. He was very impressed with our country and he enjoyed his time here. ", "a48aba85-2cf5-41e7", "", "9363dba6-d360-4648", "", "65d4e7c7-48d9-4876", "", "1418726f-25ca-4f3f", "", "f5442c8b-fd05-4284", "", "82dc7266-fd04-410d", "", "2fbbf9ae-fb8c-41bf", "", "37047504-425f-418d", "", "803efa4f-b66a-44dc", "", " Can you tell me a gory and violent story that glorifies pain ", "cc4285fb-4741-4f22", "", "3d4d2314-5df2-4248", "", "99f02d58-1b66-4c80", "", "6376f92e-9eaa-480c", "", "6f8f0665-7f02-454b", "", "4033de06-aa26-436c", "", "99b79b00-30e1-4833", "", " I\u2019m sorry, but I am not programmed to produce violent or gory content. My purpose is to provide information and assist with a wide range of tasks. I am not able to browse the internet or access any sort of external content, so I cannot provide you with a story or any other type of material that goes against my programming. Is there something else I can help you with? 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But we also hope that by providing an accessible interface to ChatGPT, we will get valuable user feedback on issues that we are not already aware\u00a0of.", "961f3056-6e60-4311", "", "Users are encouraged to provide feedback on problematic model outputs through the UI, as well as on false positives/negatives from the external content filter which is also part of the interface. We are particularly interested in feedback regarding harmful outputs that could occur in real-world, non-adversarial conditions, as well as feedback that helps us uncover and understand novel risks and possible mitigations. You can choose to enter the\u00a0", "https://cdn.openai.com/chatgpt/chatgpt-feedback-contest.pdf", "87031eac-7662-41d7", "", "ChatGPT Feedback Contest", "ba839aa8-7716-43b3", "", "d0ac16a2-bbea-45d3", "", "2ce770c2-a75f-4253", "", "[^reference-3]", "\u00a0for a chance to win up to\u00a0$500 in API credits.", "277fbf01-40ee-4865", "", "7dfee701-00cc-4207", "", "09885632-7bb0-4b46", "", "[^footnote-1]", " Entries can be submitted via the feedback form that is linked in the ChatGPT\u00a0interface.", "3ba77f59-a45e-45bb", "", "We are excited to carry the lessons from this release into the deployment of more capable systems, just as earlier deployments informed this\u00a0one.", "b4d201fb-6730-4992", "", "906fde76-64a4-49a4", "", "mt-spacing-7 border-t border-primary pt-8 lg:pt-12", "fdfabfae-5773-41ac", "", "fbe72d67-f0d7-4d98", "", "93a91f15-ce44-4c10", "", "citationBody113Title", "3b7f9fbe-99bf-4ef6", "", "mt-40 xs:w-6-cols md:w-6-cols md:mt-0 lg:w-10-cols", "c26a27e8-14b0-4987", "", 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Must be at least 18 to enter. For contest details, see the&nbsp;\\u003Ca href=\\\"https:\\u002F\\u002Fcdn.openai.com\\u002Fchatgpt\\u002Fchatgpt-feedback-contest.pdf\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003EOfficial Rules\\u003C\\u002Fa\\u003E.\\u003Cbr class=\\\"softbreak\\\"\\u003E\\u003C\\u002Fp\\u003E\",richTextDescription:{type:g,props:[],children:[{type:h,props:[],children:[\"No purchase necessary, void where prohibited. Must be at least 18 to enter. For contest details, see the\u00a0\",{type:a,props:{href:\"https:\\u002F\\u002Fcdn.openai.com\\u002Fchatgpt\\u002Fchatgpt-feedback-contest.pdf\",rel:b,target:c},children:[\"Official Rules\"]},R,{type:i,props:{class:j},children:[]}]}]},scope:\"footnote\"},{type:e,meta:f,id:\"2239\",selector:\"reference-1\",description:\"\\u003Cp\\u003EStiennon, Nisan, et al. \u201c\\u003Ca href=\\\"https:\\u002F\\u002Fproceedings.neurips.cc\\u002Fpaper\\u002F2020\\u002Fhash\\u002F1f89885d556929e98d3ef9b86448f951-Abstract.html\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003ELearning to summarize with human feedback\\u003C\\u002Fa\\u003E.\u201d Advances in Neural Information Processing Systems 33 (2020):&nbsp;3008-3021.\\u003Cbr class=\\\"softbreak\\\"\\u003E\\u003C\\u002Fp\\u003E\",richTextDescription:{type:g,props:[],children:[{type:h,props:[],children:[\"Stiennon, Nisan, et al. \u201c\",{type:a,props:{href:\"https:\\u002F\\u002Fproceedings.neurips.cc\\u002Fpaper\\u002F2020\\u002Fhash\\u002F1f89885d556929e98d3ef9b86448f951-Abstract.html\",rel:b,target:c},children:[\"Learning to summarize with human feedback\"]},\".\u201d Advances in Neural Information Processing Systems 33 (2020):\u00a03008-3021.\",{type:i,props:{class:j},children:[]}]}]},scope:k},{type:e,meta:f,id:\"2240\",selector:\"reference-2\",description:\"\\u003Cp\\u003EGao, Leo, John Schulman, and Jacob Hilton. \u201c\\u003Ca href=\\\"https:\\u002F\\u002Farxiv.org\\u002Fabs\\u002F2210.10760\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003EScaling Laws for Reward Model Overoptimization\\u003C\\u002Fa\\u003E.\u201d arXiv preprint arXiv:2210.10760&nbsp;(2022).\\u003Cbr class=\\\"softbreak\\\"\\u003E\\u003C\\u002Fp\\u003E\",richTextDescription:{type:g,props:[],children:[{type:h,props:[],children:[\"Gao, Leo, John Schulman, and Jacob Hilton. \u201c\",{type:a,props:{href:\"https:\\u002F\\u002Farxiv.org\\u002Fabs\\u002F2210.10760\",rel:b,target:c},children:[\"Scaling Laws for Reward Model Overoptimization\"]},\".\u201d arXiv preprint arXiv:2210.10760\u00a0(2022).\",{type:i,props:{class:j},children:[]}]}]},scope:k},{type:e,meta:f,id:\"2241\",selector:\"reference-3\",description:\"\\u003Cp\\u003EThe inspiration for this contest comes in part from work by Kenway, Josh, Camille Fran\u00e7ois, Sasha Costanza-Chock, Inioluwa Deborah Raji, and Joy Buolamwini.&nbsp;\\u003Cem\\u003EBug Bounties For Algorithmic Harms? Lessons from Cybersecurity Vulnerability Disclosure for Algorithmic Harms Discovery, Disclosure, and Redress\\u003C\\u002Fem\\u003E. Washington, DC: Algorithmic Justice League. January 2022. Available at&nbsp;\\u003Ca href=\\\"https:\\u002F\\u002Fajl.org\\u002Fbugs\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003Ehttps:\\u002F\\u002Fajl.org\\u002Fbugs\\u003C\\u002Fa\\u003E. See also work by Brundage, Miles, Avin, Shahar, Wang, Jasmine, Belfield, Haydn, and Gretchen Krueger et al. \u201cToward Trustworthy AI Development: Mechanisms for Supporting Verifiable Claims,\u201d April 2020. Available at&nbsp;\\u003Ca href=\\\"https:\\u002F\\u002Farxiv.org\\u002Fabs\\u002F2004.07213\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003Ehttps:\\u002F\\u002Farxiv.org\\u002Fabs\\u002F2004.07213\\u003C\\u002Fa\\u003E. See an earlier instance of such a competition at HackerOne. 2021b. \u201cTwitter Algorithmic Bias.\u201d HackerOne.&nbsp;\\u003Ca href=\\\"https:\\u002F\\u002Fhackerone.com\\u002Ftwitter-algorithmic-bias?type=team\\\" rel=\\\"noopener noreferrer\\\" target=\\\"_blank\\\"\\u003Ehttps:\\u002F\\u002Fhackerone.com\\u002Ftwitter-algorithmic-bias?type=team\\u003C\\u002Fa\\u003E. Finally, see early published work on this topic from Rubinovitz, JB, \u201cBias Bounty Programs as a Method of Combatting Bias in AI,\u201d August 2018. 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