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-   -   "New" primality test/check (https://www.mersenneforum.org/showthread.php?t=22838)

 gophne 2018-03-12 19:58

Hi CRGreathouse

Thanx for the encouragement.

I hope I can do justice to your positivity. I shall try to formulate my attempted "proof" carefully and mathematically as sound as possible. I will be delighted if the "proof" has any merit if somebody with more math knowledge and skills could take it the next level, but I am getting ahead of myself.

I will be posting on this thread as I would not want to compromise the integrity and reputation of the forum by possibly posting a worthless, clueless attempted proof. If the proof should have any merit after analysis by the normal commentators, it would garner a life of its own I am sure.

As for the primality test (for mersenne primes) it is formulaic in nature, so it is like proving the lemma for the lower mersennes, and then extending it to the rest of the set of mersenne numbers. My problem being that the "primality" check could well be too unwieldy at very high numbers. I do not have a good understanding of algorithm complexity and computation time concepts.

I shall try my very best to validate whatever I post with due respect to the readers.

 science_man_88 2018-03-12 20:03

[QUOTE=gophne;482175]Thanx I will look at the links.[/QUOTE]

[url]https://en.m.wikipedia.org/wiki/List_of_mathematical_jargon[/url]

[url]https://en.m.wikipedia.org/wiki/Mathematical_proof[/url]

[url]https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols[/url]

[url]https://en.m.wikipedia.org/wiki/List_of_mathematical_symbols[/url] may as well.

 CRGreathouse 2018-03-12 21:59

[QUOTE=gophne;482176]As for the primality test (for mersenne primes) it is formulaic in nature, so it is like proving the lemma for the lower mersennes, and then extending it to the rest of the set of mersenne numbers. My problem being that the "primality" check could well be too unwieldy at very high numbers. I do not have a good understanding of algorithm complexity and computation time concepts.[/QUOTE]

Think about it in four stages.

1. Define it precisely and unambiguously. Until you do this, you [url=https://www.theguardian.com/science/2005/sep/19/ideas.g2]can't even be wrong[/url].
2. Prove that your test holds for all primes (of the appropriate form). Once you do this you have a pseudoprimality test (but its value is still unclear).
3. Prove that your test fails to hold for all composites (of the appropriate form). Once you do this you have a primality test (but possibly a slow one).
4. Prove that your test runs in a (small) certain amount of time based on the size of the input and perhaps other characteristics.

There are in-between stages. Before you get to 3 you can perform random testing to show that your test fails for either all composites tested or a very high percentage. Before you get to 4 you can use heuristics or implementations to argue that it will be fast. Really you need to get to at least stage 2 before any of this matters though.

You don't need to get to 4 to be of value. BPSW is at 2, and Miller-Rabin is stuck there permanently. ECPP looks like it will be at 3 for the foreseeable future.

 chalsall 2018-03-12 22:47

[QUOTE=CRGreathouse;482186]Define it precisely and unambiguously. Until you do this, you [url=https://www.theguardian.com/science/2005/sep/19/ideas.g2]can't even be wrong[/url].[/QUOTE]

Along these lines, a very important thing to point out... It's perfectly OK to be wrong around here. It's a tough crowd, but being comfortable making mistakes is useful in making progress.

Some have run away from this forum because they thought they were being personally attacked. New ideas are welcomed, but are also open to deep analysis, and will be called stupid if they are. Nothing personal; deal with it.

It's a bit like the game of ruby, where you try to physically maim your opponent on the field, and then afterwards go and buy the guy a beer and laugh. Or the game of Go, where it's all fun and games until someone loses an eye.

 gophne 2018-03-13 03:51

Noted, thank you.

 gophne 2018-03-13 04:01

Agreed.

I have always accepted the critisisms I encountered on the forum in good faith. If I can also make use of a sporting analogy, it is like, as a novice tennis player, taking on Jimmy Connors, with Jimmy Connors sometimes feeling like hitting you on the head with the ball, for your audacity, and rightly so, if you cannot even get the ball to go consistently into the service court.

You just have to hope that you do not come up against a John McEnroe, who will likely bash the ball boy over the head with his racquet, due to your doltishness!

 CRGreathouse 2018-03-13 04:04

[QUOTE=chalsall;482192]Along these lines, a very important thing to point out... It's perfectly OK to be wrong around here. It's a tough crowd, but being comfortable making mistakes is useful in making progress.[/QUOTE]

Yes! :smile:

We all make mistakes, and it's good to point out mistakes you see so they can be corrected.

 gophne 2018-03-17 02:54

Dedicated to my daugter, Taryn

I would like to make my attempt at a proof of the [I]Twin Prime Conjecture[/I] on the 17/04/2018, which would be my daughter's birthday, hoping to make up for the bad days.

 gophne 2018-03-18 08:31

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate."

Leonard Euler, in G. Simmons, Calculus Gems, McGraw-Hill, New York, 1992

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"Prime numbers have always fascinated mathematicians, professional and amateur alike. They appear among the integers, seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little out of reach."

Underwood Dudley, Elementary Number Theory (Freeman, 1978) p.163

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"Who would have imagined that something as straightforward as the natural numbers (1, 2, 3, 4,...) could give birth to anything so baffling as the prime numbers (2, 3 ,5, 7, 11, ...)?"

Ian Stewart, "Jumping Champions", Scientific American, December 2000

 ATH 2018-03-18 16:18

[QUOTE=gophne;482576]I would like to make my attempt at a proof of the [I]Twin Prime Conjecture[/I] on the 17/04/2018, which would be my daughter's birthday, hoping to make up for the bad days.[/QUOTE]

Considering how many great minds have worked on this for hundreds of years, most recently Professor Terence Tao, and Dr. James Maynard: