# 77

This number is a composite.

77! + 1 is prime!

77 contains 'Lucky 7 and 11' as prime factors. [Sladcik]

The square of 77 is 5929, the concatenation of two primes, 59 and 29. [Trotter]

The concatenation of all palindromes from one up to 77 is prime. [De Geest]

The number 7^7+77^7+777^7+2 is prime with digital sum 77. [Patterson]

77 is equal to the sum of the first 8 primes. Note it is also the product of the middle two numbers of this sequence (11*7 = 77). [Kazgan]

77^77*78^78-1 is the only non-titanic prime of form a^a*b^b-1, where a, b are successive numbers that are both composite. [Loungrides]

Prime numbers that end with "77" occur more often than any other 2-digit ending among the first one million primes. [Gaydos]

There are 77 primes that never enter the repdigit sequence 1, 31, 331, 3331, 33331, etc. as factors. The sequence has general n-th term s(n) = (10^n-7)/3 and the primes correspond to n = 3, 5, 7, 11, 13, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 101, 103, 107, 127, 137, 139, 151, 157, 163, 173, 191, 211, 239, 241, 251, 271, 277, 281, 293, 317, 347, 349, 359, 397, 409, 443, 449, 457, 521, 547, 599, 601, 613, 617, 631, 641, 643, 677, 683, 691, 733, 739, 751, 757, 761, 769, 773, 797, 827, 853, 859, 881, 883, 907, 911, 919, 929, 947, 967, 991 and 997. [Schiffman]

77 is probably the maximal quantity of integers less than 100 that can be arranged in a row such that each is a multiple or a divisor of its neighbors. The solutions is not unique. Here is one due to Dmitry Kamanetsky: 62 31 93 1 57 19 76 38 2 52 26 13 39 78 6 54 27 81 9 63 21 42 84 28 56 14 98 49 7 77 11 99 33 66 22 44 88 8 48 12 36 18 72 24 96 32 64 16 80 40 20 100 50 25 75 15 45 90 30 60 10 70 35 5 85 17 34 68 4 92 46 23 69 3 87 29 58. Missing: 37 41 43 47 51 53 55 59 61 65 67 71 73 74 79 82 83 86 89 91 94 95 97. Can you find a better solution, using 78 or more terms? [Rivera]

There are 77 minimal non-single-digit primes, i.e., primes with greater than or equal to two digits such that deleting any number of its digits never gives another prime with greater than or equal to two digits. [Xayah]

The sum of the proper divisors of 77 equals 19 and the sum of primes up to 19 equals 77. Does this ever happen again? [Sariyar]