Discussion
If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as
Answer: A
Such a prime can be expressed as a sum of squares of two natural numbers.
This is Fermat's theorem on sums of two squares: an odd prime p can be written as \(a^{2} + b^{2}\) exactly when \(p \equiv 1 \pmod 4\). For example \(5 = 1^{2} + 2^{2}\), \(13 = 2^{2} + 3^{2}\), \(17 = 1^{2} + 4^{2}\), \(29 = 2^{2} + 5^{2}\).
Sums of cubes fail at once: 5 and 13 are not sums of two positive cubes (1, 8, 27 ...). Sums of square roots or cube roots of natural numbers are generally irrational and have no link with the remainder on division by 4.
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