Question

Determine all prime numbers a,b and c for which the expression x^2+b^2+c^2-1
is a perfect square

Answer 1

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Answer 2

Given:

Given expression is a² + b² + c² -1 is a perfect square.

To find:

Prime number a, b and c

Solution:

To find the integer solution we have ,

  p₁² + p₂² + p₃² -1 = k²

here p = primes

We have to consider the equation modulo 8 to calculate this as the square modulo 8 are restricted to 0, 1, 4.

according to modulo 8, all odd primes square to 1 and 2² = 4.

now, going through all the possibilities we have valid possibility which are-

      4 + 4 + 1 -1 =0 (mod 8 )

which means p₁ = p₂ = 2 and p₃ = odd

so, now original expression can be written as

2² + 2² + p₃² -1 = k²

              k² - p₃² = 7

(k + p₃) (k - p₃) = 7

therefore,

k-p₃ = 1

k + p₃ = 7

values of k and p₃ are

k = 4 and p₃ = 3

Thus the only solution is

(p₁,p₂,p₃) = (2,2,3)

so, a, b and c are 2, 2, 3 respectively.