Mathematicians Solve 'Twin Prime Conjecture' — In an Alternate Universe

livescience.com ^ | 10/29/2019 | Rafi Letzler

[BenLurkin]

"Twin primes" are primes that are two steps apart from each other on that line: 3 and 5, 5 and 7, 29 and 31, 137 and 139, and so on. The twin prime conjecture states that there are infinitely many twin primes, and that you'll keep encountering them no matter how far down the number line you go. It also states that there are infinitely many twin primes, and that you'll keep encountering them no matter how far down the number line you go. It also states that there are infinitely many prime pairs with every other possible gap between them (prime pairs that are four steps apart, eight steps apart, 200,000 steps apart, etc.). Mathematicians are pretty sure this is true.

Will Sawin of Columbia University and Mark Shusterman of the University of Wisconsin proved a version of the twin prime conjecture for the alternative universe of "finite fields": number systems that don't go to infinity like the number line, but instead loop back on themselves. You probably encounter a finite field every day on the face of a clock. It goes 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and then loops back around to 1. In that finite field, 3+3 still equals 6. But 3+11=2.

Finite fields have polynomials, or expressions like "4x" or "3x+17x^2-4," Sawin told Live Science, just like regular numbers do. Mathematicians, he said, have learned that polynomials over finite fields behave a lot like integers — the whole numbers on the number line.

Statements that are true about integers tend to also be trust about polynomials over finite fields, and vice-versa. And just like prime numbers come in pairs, polynomials come in pairs. And ... unlike integers, when you plot them on a graph they make geometric shapes.

[null and void]

W.C. Fields nephew?

Nope. That’s Phineas Fields.

Chester.

It took a while for Lucky Strike, his sponsor, to figure out that the sudden appearance of Chester Fields wasn't helping their sales...

[Gamecock]

To harness that kind of mental ability their girlfriends must have had mono for quite some time.

[cgbg]

The evil of Congressional Democrats is infinite—and that is in our dimension!

[scrabblehack]

The clock example is regular integer modulo 12.
So the prime numbers would be 2, 3, 5, 7, 11.

Now if you say 2, 3, 5, 7, 11 are primes on cycle after cycle of course there are infinitely many twin primes: (3,5) and (5,7) on every cycle.

But let’s see:
13 mod 12 = 1 (1 on the first cycle would not be a prime but it would be on the 2nd cycle and thereafter?)

17 mod 12 = 5
19 mod 12 = 7
23 mod 12 = 11

It would seem all primes (larger than 2) mod 12 would be odd numbers - I guess the problem is if one of them is a 9.

[Revel]

” The twin prime conjecture states that there are infinitely many twin primes, and that you’ll keep encountering them no matter how far down the number line you go. It also states that there are infinitely many twin primes, and that you’ll keep encountering them no matter how far down the number line you go.”

Now that is what I call a word prime. Do you know how many times I had to read that in order to convince myself that I was not seeing things?

[scrabblehack]

Of course it can’t be a 9. 9 + 12x is always divisible by 3.
It can’t be a 3 either.

[SunkenCiv]

Thanks BenLurkin.

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[SunkenCiv]

The First 100000 Twin Primes
https://primes.utm.edu/lists/small/100ktwins.txt

List of Twin primes: 1 - 1000
https://prime-numbers.info/list/twin-primes

Twin Primes — from Wolfram MathWorld
https://mathworld.wolfram.com/TwinPrimes.html

All Twin Primes under 1,000,000,000
https://www.hugin.com.au/prime/twin.php

[trebb]

Do they use these mind games to stay sharp or is there an application?