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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 starting with one sample | Comment 1 of 14

It looks like yes, n=56 creates 169 and 225 ,both perfect squares.
Evaluating members  manually of two series  a1=3*m*56+1 and a2=4*m*56+1 ,I did not find any m number causing a perfect square number in both series.

That is not a proof ,but unless a counterexample is found  I
believe that N=56 might be  the only case and therefore the following statvement is true: 

If N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square-than N is divisible by 56.

...still waiting  for more cases or counterexample...

Edited on April 2, 2010, 4:36 am
  Posted by Ady TZIDON on 2010-04-01 11:41:27

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