# Sums of squares of primes [closed]

Question: What is the least number that is a sum of three squares of primes in exactly six ways?

... I know it is not research mathematics. Happy new year!

EDIT: Now that it is answered I should note that I learned this puzzle from a tweet by Ed Southall. I thought it is fun to share.

## closed as off-topic by Wojowu, Ben Linowitz, Ben Barber, Chris Godsil, Neil HoffmanJan 5 at 1:35

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Wojowu, Neil Hoffman
• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ben Linowitz, Ben Barber, Chris Godsil
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• If you know it is not research mathematics, why post it here? Mathematics SE exists. – Wojowu Jan 4 at 12:25
• this is more like a puzzle, right? the answer is 2019 $$2019=a^2+b^2+c^2,\;\;\text{with}\;\;(a,b,c)\in\{(7,11,43),(7,17,41),(13,13,41),(11,23,37),(17,19,37),(23,23,31)\}$$ – Carlo Beenakker Jan 4 at 12:43
• @CarloBeenakker, you are right. – Andreas Thom Jan 4 at 12:47
• Is there an elegant or intuitive solution? Of course, one can easily find the answer with a computer program. – MathematicsStudent1122 Jan 4 at 17:35

$$2019=a^2+b^2+c^2,\;\;\text{with}\;\;(a,b,c)\in\{(7,11,43),(7,17,41),(13,13,41),(11,23,37),(17,19,37),(23,23,31)\}.$$
The sequence $$f(n)$$ given by those smallest integers which can be written in $$n$$ ways as the sum of squares of three primes has first eleven values,
$$12,219,363,699,1179,2019,2259,3891,4059,6459,5379.$$
As one can see, the $$10$$th value, $$6459$$, is larger than the $$11$$th value, $$5379$$. This is OEIS A214512, which indicates that it was really T.D. Noe who observed this earlier.