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1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2 n + 1 for n ≥ 0, now known as Fermat numbers. If F n happens to be prime, F n is called a Fermat prime. Fermat showed that F n is prime for each n ≤ 4, and he conjectured that F n is prime for all n (see Brown [1] or Burton [2, p. 271]). Almost one hundred years(More)
Let D be a list of single digits, and let k be a positive integer. We construct an infinite sequence of positive integers by repeatedly appending, in order, one at a time, the digits from the list D to the integer k, in one of four ways: always on the left, always on the right, alternating and starting on the left, or alternating and starting on the right.(More)
A classical theorem in number theory due to Euler states that a positive integer z can be written as the sum of two squares if and only if all prime factors q of z, with q ≡ 3 (mod 4), occur with even exponent in the prime factorization of z. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the(More)