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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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solution | Comment 1 of 3
The task is to pick n so that a^2 = b^2 + n has k solutions for a given k when a and b range over the positive integers.

Looking at (a + b)(a - b) = n, all that is needed is to pick n big enough.  

Powers of k suggest themselves.  n=k^k isn't big enough and n=k^2k fails for even k, but n=k^3k allows at least k matches.   

  Posted by xdog on 2018-12-31 08:48:09
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