# 42

This number is a composite. The sum of the least consecutive primes ending in "1" (42 = 11 + 31). [Rivera] 42^2 is the smallest perfect square that is the mean of cubed twin primes. Curiosly, the difference of the twin prime cubed and the mean of the cubed twin primes is again a prime: 42^2 = 11^3 + 433 = 13^3 - 433. [Vrba] The 13th case of a prime concatenation of the form n & n-th prime is with n = 42 and n-th prime = 181 (42181). Coincidentally, 181 is congruent to 13 modulo 42. Note that 18142181 is also prime. Furthermore the operations 181 - 42 and 42 + 181 yield primes resp. 139 and 223 leading us to these prime combinations with 42 in the middle : 13942223 (A) and 22342139 (B). Finally (B)(A) = 2234213913942223 is prime as well. [De Geest] In the first book of his Hitchhiker "Trilogy" Douglas Adams writes, "42 is the answer to life, the universe, and everything." Note that 42 lies between a twin prime pair. [Little] 42 = (π(4)*π(2)) * (prime(4)*prime(2)). Note that 42 is the only number less than 20000000 with this property. [Firoozbakht] 42^42 + π(42) is prime. It is interesting that all numbers less than 42 with this property are are all prime factors of 42. [Firoozbakht] All five numbers 1 + 42, 1 + 4242, 1 + 424242, 1 + 42424242 and 1 + 4242424242 are prime. [Firoozbakht] The smallest integer that forms a pair of twin primes by concatenation with the previous and the next number (4241, 4243). [Silva] The smallest sphenic number that is not a primorial. [Merickel] 42n+n^2+(n-1)^2 generates at least 50% prime numbers up to 20,000,000 and up to n=3160 at least 50% of the numbers are prime for all positive values of n. There are 85 primes in the first 100 positve values of n. [Homewood] If p is an odd prime, there are exactly 42 nilpotent Lie rings which have immediate descendants of order p6. [Gudipati] 42 is the smallest number k that is equal to the sum of the nonprime proper divisors of k, i.e., 42 = 1 + 6 + 14 + 21. Integers with this property are the product of a perfect number and a Carol prime. [Honaker] The determinant of the matrix formed by the four consecutive primes: 29, 31, 37, 41. [Ahmed] 42 divides the sum of digits of the first 42 prime numbers, i.e., 42 divides (2 + 3 + 5 + 7 + 1+1 + … + 1+7+9 + 1+8+1) or 378. Can you find the first prime with this property? See PC71. [Honaker] There are 42 integer partitions of the number 10. [Ahmed]

(There is one curio for this number that has not yet been approved by an editor.)