{"created_at":1790148062,"completed_at":1790148077,"id":"b0e783ec-038b-9fbf-9b30-27c804efc975","max_output_tokens":null,"model":"grok-4.7","object":"response","output":[{"id":"rs_b0e783ec-038b-9fbf-9b30-27c804efc975","summary":[{"text":"The problem asks to find all prime numbers \\( p \\) such that \\( p^2 + 2 \\) is also prime, and prove the answer.\nThe only prime \\( p \\) such that \\( p^2 + 2 \\) is also prime is \\( p = 3 \\).\n\nTo prove this, first check the smallest primes directly.","type":"summary_text"}],"type":"reasoning","status":"completed","encrypted_content":"5fJROk26jMaee6HGJ/epv3QP9zmGFZuhNlk9SENyNBxaHT0FPxhJATCC2OAB4p7KosFKjy2E5l5kePPSzgGJT9u67HY1nshLNUEVelpPJw3AiinDg4501t/YnYl8KmnriKb+OABWKUW28hwfIcxcxnu7GIk1hjYV+uPtwr7yWlUXNRDj256Z0VqfHzPzXoF+Y/TZXePG70BW/z7VwsLQtUujCnv2U1c4pFikWdJm6CELProitFvKFocRxUYELa6AAN6xI2Ly3ArTh/piTtcuIMeezGjaOnpizkgvoI7NVrhWV96s04SWpyT+i0nTfPXKbKCf5Bay5/bHEgQEKqeTB/ysOZMNye/K3WuxiUMMHYjCNS0iLsTL8DJU7Wwu1evMX2VzKiT//ZNN5TQwJYVZA6NgXoQSjANry3shEODiOi7ugvrjEGIiKQv5rfw04QMt2K8mx+BNjIno9p7IXpHtQ2ggw3fIwTEFrzsRA9xy8fhL2EPtAsAmVnNms+t8k+Bb4DzbalVWSi+/gewK6F+dQsXRjU/+neUE8lmmhTe5xN9h+BgJ0iJiizXwxNru/0dVzN81YZAx7t9Yx/p5nksXQbHyHOxqSLHGWCyrNyjlJhgM40ykmvOUSAOplKdjfL6np8E7UFlDvRnDlw3WcAA1PqIBuZueXg3SYdrwGC2u/4QkYjHEHLVsAqZM3HJw6oMczgwMIEZbJCfosDGiXFiRsSOjuxNkpH8wjWK7NafnADfZKL5Vi0VqI84gJ43ow05tlfaquSkLlijrXSngmkslUUqO5MHnVm9eu/j+7iE6PhlsdMGe0ZIdOCKQfuchHrOqQUnPqVqAgSHuUASIVcIq+23I9HtTV/nsEcvqHE3qCoQHjGc7hiM9YjftCbMu2b8LRvPmYzK37Q5hXi0Z+x8pHVS2Ksw8bO3Wy7JGKf/kZH4pqcXjUTwP9focPE88t2nZA4zj7stioRKWbU9PQaRHlBZ/uWWj1si1inPhCYbdcvqneJ1OLG8MdBzYcromroOBsxma1aKTj0whpcZXEZ0DjXLPR0Q8ZibO7Xy5znopeuWYgt8qgVCZms9C8d5Y3v2GMVw5UGYGrHqH9BpjEWiUY9k0YZXvsbgPVHNOmkSIQsBCl8bZyezsEGOzgChLBy3AjE5DH56S2kQBaXB8pn6Adbt6bVQEaGDGVxFBp9gUrBdzq+LFuyecv9t9GlZb+8ViB3euQeLRLI8WVVRqYBoWepRjSL9zzoxUTibBghCVP7wG+8OLJZEyE32QNZ6iwvUhc3F0LsQ6/8/syO9TVkLwQH+aT+yvr2uYV7xx11y0pl5EUn1gp+flCBHvjEMXHvfJ0i3wIVoQkrgBycWmE4zHDl3LzG+pTaqbl0kT5A4gwXLyo74iQXMGaJUGAhsh7mM3XZVdysSwO3pYtaU30V+y0GkX7hnhXon7WXoBPOd5InCrBSatPKr3EvqghXpWJMQ+M9uNUGFBRaSH5QEl4loOejVC9kkHorcCvHHTRH0YWdGjedoGlvWGUImEAajO6b88CBwU+/ObMxX2QgDL/0PLzOTO+tRN0/NfQ0POMS5pZBLQL8R/eAQeJ0RNv4fOjoW2fQgOQZaVEl5pO54W6lg8JoY+rsrrsB9RHxigGfXggeme7c2v8rCNop2EWPtb/LXBeM1d46t6gdHnegfQNLqbGQC4NT/6pKVHu+BpLNpwsJ0X9RCmZlh3EKP+2dxLkJ0Yvb8TpzPCnna53KlmJnZMkJ06WCz7My04ISP7WWB9XQdIQlNnnYErTVVzoc4uQ4pDaqdqw6sVCkx0g9kqC2aXw++RVMUmFvCkLNUEa+sujqzsGSN2rdt4V/uE7BQ1yUrtDCUkx8bHiLCQ/OSKUIh+mAE+qmiGsw2qrEZS5YHGBuQTNQuODpj0UWYysHb/+jSosz/RXTl5kgwxhxJ2a66TB5BbtIQbOhpxhYeeo1bue9P8UZn16Y5z/kTndaa9xzoQymMbsSzZf9BicflQwC1z3uYMjhM8DpDlH2UAySzfr7CfjQQsylJj/3jAzW3sfFa8KhRpGOiV0ycXwe4ojqK23SJtOt01EiKY/oSvVmkwW1HOVUmEAYMkOPG/tfv+IMKlBrQZr/IKqv5Zlvfp3RXoRiNr1FWLrbttv4mAzzK8FWyCQGEIZF7u2BsGFdDkf8ZvDBYdiUA8E7I6XcWCxyDJe9pI+V/Ko8xel0L9D/nZAKcPvUVwWWX8OKxv/IaFvi4cY5ruHh4JzfL+hsrsZLiiH1+IN4XifYb4SlSbXamLh0wdA6kGmiAujR858aywcaquElLG1PzJAfUmRcNq0yXAx9cvL3ZhFxUa3xED8w31Y/DZVneXpJpp+KJXuoMw1jJ3VnY7jOyBd4OJuUlInIEeeMSrakWjV2G+R2e/5pYGd6BBRKi6p85d+6LsgNwb79oTiucW7imilg2mg4+79iXuInkriFuif7KvnSf1RUAVP0hpTE9gTl+JjuLLRvh+t8wClVd9XiolwIwjn8S99iwxQtumgIa8UbpFSICIV902IcsSgqrzPQTMSYAGyLyyqHrK+pGciTSK0MqnmJwzl0qJidQwcHmwfacu8CKZ8enIeaQgkroo+KEQyRStZNj8q6wqyRr3/GkwPLh9MRt7aA1hqCzEdDXn8q7Hfh8p3kCJwiITSOGv+P+pr4ZJlHPA9JNRXXFDvWxFnMdo1eCFYgC39bpHj8+LW3G6+1kp2Ydb4l9G2+vZTOHeSIM0EWwXK6KvY52Nh63cCJysk1LyBlLzFY4DWeIAljpIUPKv0EfQ5hzeVr+0igadkPVmWh8igQ6y1aU338zY8Fs5rLp5RKQEpKRrXtMHgT3OsvZqNd6znRlmIMC4AdX3hTgQYdDJhZTi04QCoTBH5KM3zDUKHgLfK6CL2aDBB4/7zDGFaNnF3N55xfQGZicUZm9ABsxOQByLTisar4kDvgGjKrGSHxjHrgrroHOOsN+xBKZ2MUH+pgsTxEHwH8FU8zMRXQ40rUtIaoyFSFzzgtyiKpF6re5WDNde2g"},{"content":[{"type":"output_text","text":"### Setup and statement.\nLet \\( p \\) be a prime number such that \\( q := p^2 + 2 \\) is likewise prime. We claim that the only such prime is \\( p = 3 \\).\n\n### The case \\( p = 2 \\).\nHere \\( q = 4 + 2 = 6 \\), which is composite.\n\n### The case \\( p = 3 \\).\nHere \\( q = 9 + 2 = 11 \\), which is prime. Thus \\( p = 3 \\) is a solution.\n\n### The case \\( p > 3 \\).\nIn this case \\( p \\) is not divisible by \\( 3 \\), hence \\( p \\equiv 1 \\pmod{3} \\) or \\( p \\equiv 2 \\pmod{3} \\). In either subcase we have \\( p^2 \\equiv 1 \\pmod{3} \\), and thus \\( q = p^2 + 2 \\equiv 0 \\pmod{3} \\). In other words, \\( 3 \\) divides \\( q \\). But since \\( p \\ge 5 \\), we have \\( q \\ge 27 > 3 \\), so \\( q \\) is composite (being divisible by \\( 3 \\) and strictly larger than \\( 3 \\)).\n\n### Conclusion.\nThe only prime \\( p \\) for which \\( p^2 + 2 \\) is likewise prime is therefore \\( p = 3 \\).","logprobs":[],"annotations":[]}],"id":"msg_b0e783ec-038b-9fbf-9b30-27c804efc975","role":"assistant","type":"message","status":"completed"}],"parallel_tool_calls":true,"previous_response_id":null,"reasoning":{"effort":"high","summary":"detailed"},"temperature":0.7,"text":{"format":{"type":"text"}},"tool_choice":"auto","tools":[],"top_p":0.95,"usage":{"input_tokens":1274,"input_tokens_details":{"cached_tokens":1152},"output_tokens":1159,"output_tokens_details":{"reasoning_tokens":858},"total_tokens":2433,"num_sources_used":0,"num_server_side_tools_used":0,"cost_in_usd_ticks":77740000,"context_details":{"input_tokens":1274,"output_tokens":1166}},"user":null,"incomplete_details":null,"status":"completed","store":true,"metadata":{"system_fingerprint":"fp_c60958d81f66f847"},"background":false,"service_tier":"default","truncation":"disabled","top_logprobs":0,"presence_penalty":0.0,"frequency_penalty":0.0,"prompt_cache_key":null,"max_tool_calls":null,"safety_identifier":null,"error":null,"instructions":null}