Square + Square = Prime?

Number Theory Level 2

Can an odd prime being expressed as the sum of the two squares? If yes, then what form of the prime should be? (k must be any integer of course!)

no 4k+3 4k+1 uncertain

敬全 钟 by 敬全 钟 , 22, Malaysia

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2 solutions

Merseinne Prime = 2^prime -1. Thus, 4k+1

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Finn Hulse
Jan 26, 2014

A couple of numbers that fit are 29 (25 plus 4), 5 (4 plus 1), and 61 (25 plus 36). All of these numbers share one thing in common: they are all one greater than a multiple of 4.

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Ya I did the same thing ....but was thinking whether it can be proved?

Varenya Srivastava - 7 years ago

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Squares are congruent to 0 0 or 1 ( m o d 4 ) 1\pmod 4 , which means the sum of two squares is congruent to 0 , 1 , 2 ( m o d 4 ) 0,1,2\pmod 4 . Since the prime is odd, it is congruent to 1 ( m o d 4 ) 1\pmod 4 .

mathh mathh - 6 years, 11 months ago

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