Prime Sum

Number Theory Level 3

Find the sum of all primes $p$ for which the system

$\qquad \qquad p+1=2x^2 \\\qquad \qquad p^2+1=2y^2$

has a solution in integers $x,y$ .

The answer is 7.

2 solutions

Rasmus Brammer
Nov 19, 2014

Rewrite: p + 1 = 2x^2 <=> p = 2x^2-1 factorize it and let a = (2^0,5)x we then get: p = (a-1)(a+1). Since one of the fractions has to be 1 and the other a prime. Since a = 2 is the only solution we need to check if it is fact is a solution.

a = 2 <=> 2 = 2^(0,5)x , x = 2. We check if this is fact is a prime:

2*2^2 -1 = 7. Now we only have to check if we can find a integer, y, which satisfy the second equation:

p^2 + 1 = 2y^2 <=> 7^2 + 1 = 2y^2. We see that y = 5 gives us a right solution. Hence the sum of all primes for which the system has a solution in integers x,y is equal to 7

Akash Deep
Nov 12, 2014

the key is to use that , x < y . its very difficult to write the solution after so much coding so just obtain a relation between x and y by the 2 equations. then
you will obtain 2x^2(x+1)(x-1) = (y+1)(y-1). euation - 1 now note that y+1 is greater than x , x+1 and x - 1. so, y+1 is not equal any of these 3 or hidden factors in them. so now y - 1 can be equal to 2x , 2x + 2 or 2x -2 because we also have to ensure that y + 1 > y -1 . on trial we obtain that y - 1 = 2x gives us a consistent solution according the other conditions. if y - 1 =2x then, y = 2x + 1 now put the value of y in equation 1 and obtain a cubic equation in x , that on solving will be x^3 + 3x - 2 = 0, we obtain only one +ve integral value for x , that is x = 2 now p = 7.

Prime Sum

The answer is 7.

2 solutions

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