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 jwaltos 2021-01-15 05:40

MArt

..short for math and art.

This is the reason behind the post:
[url]http://www.concinnitasproject.org/portfolio/[/url]

Perhaps these could be used as a "Rorschach" test.

 Nick 2021-01-15 10:54

For anyone who hasn't seen it yet:
[URL="https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK"]Proofs from The Book[/URL]

 xilman 2021-01-15 15:09

[QUOTE=Nick;569340]For anyone who hasn't seen it yet:
[URL="https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK"]Proofs from The Book[/URL][/QUOTE]I've not see the book itself so can't tell whether one of the six proofs of the infinitude of primes is the very elegant one based on the factorization of Mersenne numbers and Fermat numbers.

The basic idea is that F_n - 2 = 2^2^n - 1 = (2^2^(n-1) +1) (2^2^(n-1) -1) = F_{n-1} * (F_{n-1} -2)

by the difference of squares factorization formula and noting that F_n is co-prime to F_m when m != m.

 Nick 2021-01-15 15:37

[QUOTE=xilman;569364]I've not see the book itself so can't tell whether one of the six proofs of the infinitude of primes is the very elegant one based on the factorization of Mersenne numbers and Fermat numbers.

The basic idea is that F_n - 2 = 2^2^n - 1 = (2^2^(n-1) +1) (2^2^(n-1) -1) = F_{n-1} * (F_{n-1} -2)

by the difference of squares factorization formula and noting that F_n is co-prime to F_m when m != m.[/QUOTE]
Yes, that's the 2nd one.
And the 3rd one uses Mersenne numbers (for prime p, a prime factor of \(M_p\) is greater than p).

 petrw1 2021-01-15 16:32

[QUOTE=jwaltos;569332]..short for math and art.
[/QUOTE]

Why do I only see a dog watching football and jumping off the couch?

 jwaltos 2021-01-16 05:52

[QUOTE=petrw1;569376]Why do I only see a dog watching football and jumping off the couch?[/QUOTE]

Only?

 jwaltos 2021-01-16 06:05

[QUOTE=Nick;569340]For anyone who hasn't seen it yet:
[URL="https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK"]Proofs from The Book[/URL][/QUOTE]
The 6th edition regarding the "Sums of two squares" can be followed up with:[url]https://en.wikipedia.org/wiki/Geometry_and_the_Imagination[/url] (p.32, 1990 translation);[url]https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat;[/url];[url]https://www.cambridge.org/core/books/number-theory-in-the-spirit-of-liouville/51A0D57710C50412C1C535049FACCE33[/url].
There is more to this than meets the eye.

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