{"schema":"vela.problem-packet.v0.1","problem":479,"statement":"Is it true that, for all $k\\neq 1$, there are infinitely many $n$ such that $2^n\\equiv k\\pmod{n}$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A006517","name":"Numbers k such that k divides 2^k + 2.","terms":"1,2,6,66,946,8646,180246,199606,265826,383846,1234806,3757426,9880278,14304466,23612226,27052806,43091686,63265474,66154","url":"https://oeis.org/A006517"},{"id":"A006521","name":"Numbers n such that n divides 2^n + 1.","terms":"1,3,9,27,81,171,243,513,729,1539,2187,3249,4617,6561,9747,13203,13851,19683,29241,39609,41553,59049,61731,87723,97641,11","url":"https://oeis.org/A006521"},{"id":"A015919","name":"Positive integers k such that 2^k == 2 (mod k).","terms":"1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157","url":"https://oeis.org/A015919"},{"id":"A015921","name":"Positive integers n such that 2^n == 4 (mod n).","terms":"1,2,4,6,10,12,14,22,26,30,34,38,46,58,62,74,82,86,94,106,118,122,132,134,142,146,158,166,170,178,182,194,202,206,214,218","url":"https://oeis.org/A015921"},{"id":"A015940","name":"Positive integers n such that 2^n == -3 (mod n).","terms":"1,5,917,3223,62911,326329,395819,33504053,4446226763,17556128765,141613728437,5259417592253,113837290408523","url":"https://oeis.org/A015940"},{"id":"A036236","name":"Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.","terms":"1,0,3,4700063497,6,19147,10669,25,9,2228071,18,262279,3763,95,1010,481,20,45,35,2873,2951,3175999,42,555,50,95921,27,174","url":"https://oeis.org/A036236"},{"id":"A050259","name":"Numbers k such that 2^k == 3 (mod k).","terms":"1,4700063497,3468371109448915,8365386194032363,10991007971508067","url":"https://oeis.org/A050259"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}