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Perfect Square To Divisibility By 56 (Posted on 2010-04-01) Difficulty: 4 of 5
N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square.

Is N always divisible by 56?

If so, prove it. Otherwise, give a counterexample.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Now, four values | Comment 5 of 14 |

I ran a brute force search up to 500,000,000 and found four values:
56=56*1,
10920=56*195
2118480=56*37830
410974256=56*7338826

These values are approximated by 56*10^(2.2878*k+.0022) for k=0 to 3.  Taking k=4 suggests the next solution to be in the neighborhood of 56*10^9.1534=79,723,000,000

Edit: My test just finished, fifth value confirmed one answer (no non-multiples of 56 showed up): 79,762,887,240=56*1,423,694,415

Edited on April 1, 2010, 8:59 pm
  Posted by Brian Smith on 2010-04-01 20:55:46

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