Abstract: We study positive integers whose binary representation is a palindrome of the form (1 . . . 1)(0 . . . 0)(1 . . . 1), where the number of ones is twice the integer a ≥ 1, and the number of zeros is b ≥ 0. We give an elementary algebraic description of all such integers, derive a factorization formula, and deduce consequences for primality. In particular we show that except for the case a = 1 (which yields the numbers $$2^{m} + 1$$ with m = b + 1), every such integer is composite by a simple factorization; consequently the only possible primes in this family are Fermat-type numbers $$2^{m} + 1$$, and thus (by a classical necessary condition) their exponents m must be powers of two. Several illustrative examples are given and implications for search strategies are discussed.

Keywords: binary palindrome, repunit, Fermat number, factorization, primality, computer algebra system Maple

Pages: 100-105

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