A number theory problem by Chirayu Bhardwaj

Number Theory Level 4

The number of prime numbers less than 100 which can be expressed as the sum of the squares of two natural numbers not necessarily distinct is

13 15 12 11 10 20 16

Chirayu Bhardwaj by Chirayu Bhardwaj , 21, India

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1 solution

Chirayu Bhardwaj
Jun 12, 2016

@Brian Charlesworth,@Chew-Seong Cheong,@Otto Bretscher or anyone else please upload a solution as i am not able to solve it. please :) it will be appriciated.

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This involves an application of Fermat's Two Squares Theorem . To this end, only primes p = 2 p = 2 and those of the form p ≡ 1 ( m o d 4 ) p \equiv 1 \pmod{4} satisfy the required condition. As for the actual counting of such primes, I don't think that there is any kind of formula, so we just have to check each integer 4 n + 1 4n + 1 for 1 ≤ n ≤ 24 1 \le n \le 24 to see if it is prime, or we could go through a list of primes and see which can be expressed in the form 4 n + 1 4n + 1 . Here is a list of such primes.

Brian Charlesworth - 5 years ago

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Thank you very much sir :).

Chirayu Bhardwaj - 5 years ago

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