Talk:1105 as Sum of Two Squares

The proof I've submitted here seems to rub up against the limits of human-readability. While it's in possible for someone to read through the table and subsequent list to validate the theorem, they'd probably have an easier time simply redoing the calculation themselves by computer. Does ProofWiki have another way of handling computation intensive results like this? The common mathematical practice of simply saying "it follows by direct computation," would be one extreme, what I've done here is other. One could also imagine doing submitting a short program or pseudocode. --Wpcavendish (talk) 19:07, 1 April 2022 (UTC)

There is precedence of posting code (which I usually put in the Talk page). There also seems to be talks of constructing pseudocode.

In this current situation, a short program is obviously more informative than a demonstration of an exhaustive search. I am not sure what the limit to an exhaustive search should be (I may have violated a limit with this). However I believe we need to first make sure that there is no (neat) analytical method to narrow down our search range. (In this example we have Integer as Sum of Two Squares and Fermat's Two Squares Theorem which can exclude a large number of zeros and ones in the long table.)

Also I don't think showing that $x^2$ is increasing is necessary. Even if it is, there has to be a "Square Function is Increasing on Positive Reals" result, or, if there isn't and this result is really necessary, show that $n^2$ is an strictly increasing sequence instead. ($\paren {n + 1}^2 - n^2 = 2 n + 1 > 0$) --RandomUndergrad (talk) 20:15, 1 April 2022 (UTC)

This page reaches (in fact, IMO way bypasses) the size limit of a manageable page. There is a significant delay between pressing the key and its result appearing on the screen, for instance. Hence I support RandomUndergrad's assessment. In this case there has to be a better way than an exhaustive list. A program, from which we may then report on significant points, is the way to go.

I will call your attention to how links are presented in $\mathsf{Pr} \infty \mathsf{fWiki}$. Please diff mine with yours and see the differences. Also, note that while it may seem easy to write $\LaTeX$, like all these things it takes more effort to make it presentable. We have a rigorous house style which you are encouraged to assimilate. --prime mover (talk) 21:34, 1 April 2022 (UTC)

You explicitly show 1106 cases, but

276 $\paren {\frac 1 4 }$ of them are completely unnecessary because $n \equiv 3 \pmod 4$ will never be the sum of squares

184 $\paren {\frac 1 6 = \frac 2 9 \times \frac 3 4 }$ of them are completely unnecessary because $n = 3^1 \times k$, $3 \nmid k$ will never be the sum of squares

76 more are unnecessary because $n = 7^1 \times k$, $7 \nmid k$ will never be the sum of squares

44 more are unnecessary because $n = 11^1 \times k$, $11 \nmid k$ will never be the sum of squares

More than half of the table can be eliminated with quick/easy logic, but the optimal solution is code in R or Python or the programming language of your choice. --Robkahn131 (talk) 00:16, 2 April 2022 (UTC)

Great – thanks for your comments. I agree that code makes the most sense here. I’ve updated the page using RandomUndergraduate’s post linked above as template.

As for more elegant approaches, the best theorem I’m aware of that’s relevant here is that a natural number n can be represented as a sum of two squares if and only if each prime factor congruent to 3 mod 4 appears with an even exponent in the prime decomposition of n (this is one of the early entries in “Proofs from THE BOOK.”) But that doesn’t help much here, since it still leaves over a third of the candidates to check.--Wpcavendish (talk) 22:26, 3 April 2022 (UTC)

You're okay with us deleting that page with the table in it? --prime mover (talk) 22:31, 3 April 2022 (UTC)

Yes, I'm happy for it to be deleted. Thanks!--Wpcavendish (talk) 22:36, 3 April 2022 (UTC)

job done --prime mover (talk) 22:40, 3 April 2022 (UTC)