Sum of two squares in three distinct ways

Gwyn

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Nov 22, 2019
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Are the integers that can be expressed as the sum of two squares in three distinct ways all multiples of five?
If so why? All those that I have found using Mathematica seem to be.
 

Jomo

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Dec 30, 2014
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4,797
What do you think? What have you tried to prove this? Where are you stuck? In order to help you we need to know where you are stuck.
 

Gwyn

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Nov 22, 2019
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I have found all those for numbers below 1500
325 425 650 725 845 850 925 1025 1300 1325 1445 1450
all appear to be multiples of 5 and can be done in only three ways e.g. 325 (the smallest)
1^2 + 18^2 = 325
6^2 + 17^2 = 325
10^2 + 15^2 = 325
As far proving this is concerned I have very little idea how to start!
Was hopping someone might suggest a method or whether a proof already exists. ( I have only minor Knowledge of number Theory)
 

Gwyn

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Nov 22, 2019
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In case anyone is Interested the following disproves my Hypothesis!
2873 is not a multiple of 5 and can be done in three distinct ways
2873 = 8 ^2 + 53^2
2873 = 13 ^2 + 52^2
2873 = 32 ^2 + 43^2
 
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