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A couple of squares (Posted on 2018-02-02) Difficulty: 3 of 5
Find all integers a, b, c where c is a prime number such that
a^b + c and a^b - c are square numbers.

Source: Pedro Henrique O. Pantoja, Brazil

See The Solution Submitted by Ady TZIDON    
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solution | Comment 6 of 9 |
This simple approach works.

2c is a difference of squares so 2c=(x+y)(x-y).

With prime c, the larger factor on RHS can only equal 2c or c.

The first case implies 2x=2c+1, an impossibility.

In the second case, 2x=c+2 which is impossible for odd c, so c=2 and x=2.  Then a^b +2=4 and a=2, b=1, and y=0.  

  Posted by xdog on 2018-02-03 07:20:31
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