"I am experimenting with heuristic sieves."

[Dobri]

2¹²⁷⁷ - 1 = 1 + 2⋅3⋅5⋅23⋅59⋅89⋅233⋅397⋅683⋅1103⋅1277⋅2089⋅2113⋅18503⋅64439⋅181193⋅3033169⋅107367629⋅...
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem

[Batalov]

Originally Posted by Dobri

2¹²⁷⁷ - 1 = 1 + 2⋅3⋅5⋅23⋅59⋅89⋅233⋅397⋅683⋅1103⋅1277⋅2089⋅2113⋅18503⋅64439⋅181193⋅3033169⋅107367629⋅...
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem

Trolling much?

What does it have to do with anything?

[Dobri]

Originally Posted by Batalov

What does it have to do with anything?

I am experimenting with heuristic sieves. One of the heuristics attempts to use the known small factors of 2^p-1-1.
Interestingly, in the case of p = 1277, there is a plenty of factors of 2^p-1-1.

[Batalov]

Originally Posted by Dobri

...Interestingly, in the case of p = 1277, there is a plenty of factors of 2^p-1-1.

O RLY?
Ever heard of
x²-1 = ... ? (write it out, don't be shy)
x¹¹-1 = ... ? (write it out)

There is so much "interesting" in the world to anyone who skipped all classes in the 7th grade.

[Dobri]

Originally Posted by Batalov

There is so much "interesting" in the world to anyone who skipped all classes in the 7th grade.

Obviously, it stems from the observation that 1276 = 4⋅11⋅29. You assume the worst in others.
One should not forget to be polite and respectful to the ones who attempt to communicate instead of posting uncivilized forcible content of provocative and insulting nature as only bullies do from the assumed position of power.
At least, I remember from kindergarten the basics of being kind to people.

[Batalov]

Originally Posted by Dobri

You assume the worst in others.

You, actually, appear to be assuming the worst in others.

I almost always apply Hanlon's_razor. My remark had nothing to do with you but with your argument.

Do you disagree with an illustrative argument that if a person cut all classes in school in the past, they will later in life find a lot of things interesting and even stunning? "A kilogram of led weighs the same as a kilogram of feathers! Who could have thought?!" You are arguing not with the presented argument, but with the "assumed" fact that it was "about you". It was not.

Whether you immediately project yourself in every sentence is entirely a matter of your choice. Your assumptions.

[Dobri]

Originally Posted by Batalov

My remark had nothing to do with you but with your argument.

Same here. Only the first line of my previous response was concerned with your comment concerning the homogeneous version of the geometric sum formula.
The rest of said response was an illustrative argument.
As for razors, I do not use such tools. Sometimes, one could inadvertently 'cut' oneself while attempting to apply razors to others.

[Batalov]

I can only add another illustrative vignette -

Interestingly (c), 2¹²⁷⁸-1 is fully factored. But how can that be without the help of factoring 2¹²⁷⁸-1-1, which is not factored at all beyond the trivial factor of 2? It is one of the two: 1. Wizardry! or 2. Factoring N-1 actually helps nothing to factor N; it is a red herring.

[Dobri]

Originally Posted by Batalov

Interestingly (c), 2¹²⁷⁸-1 is fully factored. But how can that be without the help of factoring 2¹²⁷⁸-1-1, which is not factored at all beyond the trivial factor of 2? It is one of the two: 1. Wizardry! or 2. Factoring N-1 actually helps nothing to factor N; it is a red herring.

Or 3. The factors could be used indirectly in the design and testing of the heuristic sieve.
Every heuristic initially has an element of wizardry though.

[Dr Sardonicus]

Originally Posted by Dobri

I am experimenting with heuristic sieves. One of the heuristics attempts to use the known small factors of 2^p-1-1.
<snip>

A heuristic is an informed guess. Those offered seriously include the information on which they are based, what simplifying assumptions are being made, and a "plausibility argument" explaining how those principles and assumptions are being applied.

I see no indication that your "heuristic sieve" is informed by anything, or that it is anything other than phantasmagorical. You have provided no motivation for using factors of M_p - 1 in a sieve for trying to factor M_p, and have given absolutely no indication of what set, if any, you might want to use them to sieve.

You are certainly not informed on the subject by what is available on this Forum.

The question of whether factoring M_p - 1 would help in factoring M_p was discussed on this very forum about a year ago, here. It specifically mentions the exponent 1277. And, as of this posting, the thread title is still visible on the Mersenne Forum home page.

The short answer to the question is "No."

I conclude that you have either not bothered to avail yourself of the most basic facts concerning factors of M_p - or are deliberately disregarding them, in which case you are a troll.

[tuckerkao]

Originally Posted by Batalov

x²-1 = ... ? (write it out, don't be shy)
x¹¹-1 = ... ? (write it out)

x²-1 = (x - 1)(x +1)
x³-1 = (x - 1)(x² + x + 1)
x³+1 = (x + 1)(x² - x + 1)

x⁶-1 = (x³-1)(x³+1)

x¹¹-1 = didn't get any Google Search results, but since there are answers for x⁵-1 and x⁵+1, the same patterns should be observable.

x⁵-1 = (x - 1)(x⁴ + x³ + x² + x + 1)
x⁵+1 = (x + 1)(x⁴ - x³ + x² - x + 1)

x^2p+1 = Maybe a prime
x^2p-1 = Always composite
x^2p±1+1 = Always composite
x^2p±1-1 = Always composite