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Journal of Number Theory


We show that for any positive integer n, there is some fixed A such that d(x) = d(x +n) = A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x +n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x > 1is the multiset of nonzero exponents which appear in the prime factorization of x.

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