Find all positive integer n such that 2^8+2^11+2^n is perfect sq. number. Thus, find all positive integer pair (a,b) such that a^3+6ab+1 and b^3+6ab+1 are both perfect cube numbers.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of n has to be determined such that 2^8+2^11+2^n is a perfect square.

2^8+2^11+2^n

= 2304 + 2^n

= 9*256 + 2^n

= 9*2^8 + 2^n

= 2^8*(9 + 2^(n-8))

If this is a perfect square, 9 + 2^(n-8) has to be a square.

The Pythagorean triplet (3, 4, 5) gives 9 + 16 = 25

2^(n-8) = 16

=> n = 12

The expression 2^8+2^11+2^n a perfect square for n = 12

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