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2 sets with common members (Posted on 2018-12-30) Difficulty: 3 of 5
Let k be a positive integer.
Prove that there must exist a positive integer n such that the sets A( A={1^2, 2^2, 3^2, . . . }) and B ( B = {1^2+n, 2^2+n, 3^2+n, . . . }) share k identical members.

No Solution Yet Submitted by Ady TZIDON    
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Hints/Tips re: solution Comment 3 of 3 |
(In reply to solution by xdog)

For any pair of distinct odd primes p and q, the value p*q^(k-1) has exactly k distinct factorizations.

Since p and q are odd primes then their factors are all odd which then implies the sum and difference of the factor pairs will both be even.

Let f and g be a factorization pair of n=p*q^(k-1).  Then the system a+b=f and a-b=g has integer solutions for each possible factorization of n.

  Posted by Brian Smith on 2018-12-31 11:56:39
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