# 169

This number is a composite. 169 = (27 + 72) - (7 + 1) and is the smallest perfect square of the form (2p + p2) - (p + 1). [Charles] The smallest square that is prime when turned upside down. [Wu] 169 is the first composite Pell number with a prime index. Coincidentally, it is also the last square in the Pell sequence. [Axoy] The sum of first 12 powers of semiprimes is (12+1)^2. [Post] The sum of first five emirps can be represented as the square of the first emirp. [Loungrides] 169^100-168^99 is the smallest prime of form x^100-(x-1)^99. Note that 10099 is also prime and 169+168 is an emirp. [Loungrides] The lucky numbers of Euler are well known. But what will happen if we change the formula a little? Consider A = 169 - n - n^2. The expression A is prime for n = 1 to 12 and |A| is prime for n = 1 to 24. Do you know another number which give the same results except 4, 9, 25 and 49? [Petrov] The smallest square such that every digit (d) repeated d times gives prime. Note that the same is true if we reverse or invert the digits of the number 169, i.e. 1666666999999999, 9999999996666661 and 6666669999999991 are all primes. [Petrov] If A = 1, B = 2, C = 3, … , Z = 26, then PETER WALLRODT is "brilliant," just like M^2. [Worrom]

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