Are Legendre symbols proven to be defective?

[paulunderwood]

Again, you didn't quote me, but my friend claims that NewPGen spits out the number 2^3355584+1 as having a factor of 2, which is clearly impossible.

GIGO

[R.D. Silverman]

[QUOTE=jasong;128994]

Originally Posted by R.D. Silverman

I could say the same about you, just not on the subject of math. Have you ever heard of the golden rule?


I suggest you consult a life coach on personal etiquette. This has gone beyond sniping in your case, and I'm not simply talking about our encounters.

I am not politically correct. When someone spouts nonsense, I say so.
When someone exhibits clear signs of paranoia, I say so. This is a
public forum. When the town fool spouts nonsense on the street corner
he can expect to be criticized. And I wasn't criticizing YOU, I was
criticizing your friend.

I am clueless about many things. When and if I were to speak
gibberish about one of them then strong criticism would be
appropriate. But your "friend" was discussing
mathematics. And in this regard he clearly is clueless.

And I am not the only one to suggest that you need help with your
paranoia.

[jasong]

Originally Posted by paulunderwood

GIGO

lol, it was a computer program, and the number was entered as 2*2^3355583+1. Unless you're badly stating that Paul Jobling made a buggy program, GIGO most definitely does NOT apply here.

Edit: I'm not suggesting that Mr. Jobling is at fault, but I think people need to seriously consider the possibility that, when determining whether a p-value is a possible divisor, Legendre symbols aren't totally dependable. I don't have the math skills to prove it, but I have tremendous amount of respect for my unnamed friend, and that is his opinion.

While I may not look that intelligent on Mersenne Forum, and the fiasco that occurred regarding me and the No Prime Left Behind project only makes that view worse, I can assure you that my ability to reason ranks pretty high. In my opinion, the problem is that I suggest things that could be clarified in about 5 minutes of real-world conversation, but Mersenne Forum is not even close to that medium.

[R.D. Silverman]

Originally Posted by jasong

Edit: I'm not suggesting that Mr. Jobling is at fault, but I think people need to seriously consider the possibility that, when determining whether a p-value is a possible divisor, Legendre symbols aren't totally dependable. I don't have the math skills to prove it, but I have tremendous amount of respect for my unnamed friend, and that is his opinion.

Would someone else care to take a crack at determining what

"determining whether a p-value is a possible divisor, Legendre symbols aren't totally dependable"

might mean??? What does it mean for a mathematical function to be
undependable? Was this phrase formulated by someone on
Quaaludes????

[jasonp]

Originally Posted by R.D. Silverman

Would someone else care to take a crack at determining what

"determining whether a p-value is a possible divisor, Legendre symbols aren't totally dependable"

might mean??? What does it mean for a mathematical function to be
undependable? Was this phrase formulated by someone on
Quaaludes????

I think he's referring to the sieve-based precomputations for Riesel-like projects, that use big tables and legendre symbols to eliminate candidate numbers before proceeding to a proth test. Or not <shrug>

On another note, you may be the last person left who actually uses the word 'quaaludes'.

[wblipp]

Originally Posted by R.D. Silverman

This is just 1st or second year junior high school polynomial algebra. x^ab + 1 is divisible by x^a+1 when b is odd. I would have expected even Jason to know this.

You're assessment of Jason's mathematical sophistication exceeds mine. When somebody asks "can you show me a factor." I assume they don't know how to find a factor.

[ewmayer]

Originally Posted by jasong

(2) The idea that a Fermat prime cannot have odd prime factors in the exponent, is it proven, or just BELIEVED to be effective?

It can easily and rigorously be shown that 2ⁿ+1 can be prime iff n is a power of 2. End of story there.

(3) How many of the top-5000 primes have been tested via integer based math? After that twin prime fiasco with Intel, wouldn't it be better to test numbers with an integer based test after the normal test is run?

I expect very few, if any, of the ones above 10000 digits. But that is not due to some Intel-sponsored conspiracy amongst computationalists, but rather for a simple practical reason, namely that to test such large primes invariably requires many multiplies modulo same, and on modern computer hardware this is significantly faster [anywhere from 2-10x, depending on the code and the modulus] using floating-point-based convolution [i.e. FFT or some similar kind of discrete transform] than integer-based. This is not because integer multiply [and hardware multiply is the bottleneck in all such cases] is intrinsically harder to do than floating-point - quite the opposite, in fact - but rather because hardware manufacturers have thrown much more silicon at floating-point units on their chips. That's because relatively few "real world" applications need screamingly fast general-purpose integer multiply - most programs running on a typical PC need integer MUL mainly for address computations, and for the standard hardware-supported data types such address computations can instead be done using cheaper shift/add arithmetic.

As I said, there's a lot of stuff that I'm suppressing on request. My friend has told me(approximate quote),"A lot of the stuff that people accept as fact is not true, so-and-so(note: I'm not telling who or what was mentioned, but it isn't George :) ) intentionally stacked the deck so that they would find the first 10-million digit prime. Because of this, a lot of primes on the top-5000 list are actually composite. You(jasong) have the necessary skills to figure it out with a good amount of work." From the way he said it(as I said, it's an approximate quote), it sounds like people have discovered what he discovered in the past(further than you would believe) and it is being suppressed.

And you actually bought into this stereotypical crank paranoia? It's not as if projects like GIMPS operate in secret - we test prime candidates as fast as we can given the software and user base, and the prime-candidate "wavefront" advances accordingly. Sure, some projects deliberately try to winnow their prime candidates down and not bother with the lower ranges so as to get to the next artificial milestone first, but anyone is free to try their hand at this sort of thing. So pray tell [or pray ask your buddy Deep Throat]: in what way is the "deck being stacked?" If he really has a program that is 100x faster than the LL test as you've repeatedly claimed, surely he has managed thereby to stacxk the deck in his own favor?

Also, I've little doubt that at least some nonvanishing fraction of the top 5000 were incorrectly flagged as prime - any of the ones small enough to not be of much interest [i.e. most of them] and tested just once or using the same software twice without the kinds of independent-data-on-both-runs requirements GIMPS places on double checks is a candidate for bogus primehood, but "lots?" I suggest as a simple "bullshit check" to ask your friend to point out [say] 10 of the 5000 which are in fact composite. If his claim of "lots" is true, that should be easy. I'm predicting he'll come bacxk with some cryptic Yoda-like response that puts the burden of proof on you, e.g. "Trust your feelings and search diligently, you must."

Lastly, is there an integer based prime testing program that would work on an x86 LInux box? If I could get my hands on one and use it, I could establish 100% in my own mind that I wasn't simply being taken for a ride. I don't think my friend is a liar, but there are only two possibilities left, delusional and genius. I badly want to believe it's the latter, but I don't have the education to have a worthwhile opinion. Right now, it's totally up in the air.

AFAIK Jasonp's integer-based code [which uses an efficient small-prime modulus, the Mersenne M61] is the fastest of the lot. And for the record, I vote "delusional" - the patently ridiculous "will blow LL test out of the water ... 100x faster..." claims you've been making on his behalf cinch that as far as I'm concerned. On your part, you readily admit you lack the education to judge such things, yet you say that your Secret Subgenius buddy trusts you to crack the Mystery of the Stealth Composites lurking amongst the Top 5000 list. That should concern you right there - recall the old Groucho Marx quote about refusing to join any club that would have him as a member.

And please, don't ask me to provide you with links to and how-to-use instructions for jasonp's code - a simple websearch will get you there.

[ewmayer]

Originally Posted by jasonp

On another note, you may be the last person left who actually uses the word 'quaaludes'.

Yes, Bob is dating himself, but Bob has an admirably reckless disregard for fashion and fads. The rest of us, slaves to hipster trends as we are, probably would prefer more up-to-date jargon such as "Dude, put down the crack pipe!" or even more trendily, the text-message-y "d00d, R U Hi?"

[jasong]

Okay, obviously the shit has hit the fan by now, so I'll throw my own shit now.

If you tell NewPGen to sieve k=1 to 5 for the equation k*2^3355583+1, it will say everything has been sieved and all have been found composite, and it will do that in less than a second.

Given that, it should be simple for any of you braggarts to track down a factor of 2^3355584+1, and then run a doublecheck on that factor that doesn't use shortcuts. I predict that if you go through with this, you'll fail either on the first task, or the factor will be proven wrong on the second task.

If you can supply me with a factor and post it, then I will go to my friends at Riesel Sieve, who find me entertaining, if not intelligent, and ask them to duplicate the test. Because I really don't trust you guys to be honest about this, not because I'm delusional, but because a lot of people on here(I'll leave wblipp out of this, since this seems to be the first time he's been disrespectful to me, maybe some others deserve the same honor) seem to be extremely proud of their accomplishments, while my values won't even let me read self-esteem literature without getting offended.

There are a lot of jackasses on this forum, and I would venture to say that the so-called crackpots tend to deserve more respect than the people who seem to exist to degrade them.

If you have a tendency to show respect to people who don't understand number theory, my hat's off to you. Negative energy is more powerful than positive energy, which is the reason high-mindedness needs to be nurtured wherever it is found. With the things my friend told me, I have lost almost all interest in prime number projects, though wblipp may continue to get my computer time, simply because I've found him to be respectful a majority of the time.(99% it looks like, maybe more) And of course, OPN project primes are verified in a different fashion, by necessity.

I'll repeat my challenge. Find a factor for 2^3355584+1 and test it without shortcuts. If my friends(or tolerators, possibly) at Riesel Sieve tell me that they've duplicated the result, then I'll shut up. Otherwise, you should seriously consider showing so-called cranks more respect.

[mdettweiler]

Originally Posted by jasong

If you tell NewPGen to sieve k=1 to 5 for the equation k*2^3355583+1, it will say everything has been sieved and all have been found composite, and it will do that in less than a second.

Given that, it should be simple for any of you braggarts to track down a factor of 2^3355584+1, and then run a doublecheck on that factor that doesn't use shortcuts. I predict that if you go through with this, you'll fail either on the first task, or the factor will be proven wrong on the second task.

Using the latest sr1sieve, I found that 2^3355584+1 is divisible by 5. I verified the factor with Pari/GP (which as far as I know uses plain old straight division, with no "shortcuts").

[wblipp]

Originally Posted by jasong

Given that, it should be simple for any of you braggarts to track down a factor of 2^3355584+1, and then run a doublecheck on that factor that doesn't use shortcuts.

Using the same high school algebra that Dr. Silverman so generously assumed you would know - because 3355584 = 64 * 3 * 17477, it has the following algebraic factors,

2⁶⁴+1
2^64*3+1
2^64*17477+1

All can be checked with pencil and paper using numbers no larger than the exponents themselves and the previously mentioned high school algebra.

Originally Posted by jasong

then I'll shut up.

It's time to deliver on this. You should check your meds before posting again. I think you're going to be sorry about some of these posts tomorrow or the next day.

William