# 83

This number is a prime.

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The only prime of form 3^x+5^x+7^x, where x is prime, (case x=2). Note that every other number of the same form, for x a prime, ends with 5. [Loungrides]

The only non-titanic prime of form p^(p+1)+2, where p is a prime, i.e., 3^4+2 = 83. [Loungrides]

(83, 89, 97) is the larger of only three known cases of successive primes p < q < r such that p|(qr-1), i.e., the smallest prime divides the product of the other two minus 1, (83 divides 89*97-1). The previous two cases are (2, 3, 5) and (11, 13, 17). [Loungrides]

Prime numbers that end with the prime "83" occur more often than any of the possible two-digit endings of primes less than a thousand. These are 83, 283, 383, 683, 883, and 983. [Loungrides]