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Search: Posts Made By: paulunderwood
Forum: Information & Answers 2021-02-24, 22:52
Replies: 18
Views: 218
Posted By paulunderwood
if it is divisible by n it leaves a remainder of...

if it is divisible by n it leaves a remainder of 0 by division which by definition is 0 mod n. After all n divides 0.

In my example 7!/(5!*2!). Don't forget not to divide by n. We have
...
Forum: Information & Answers 2021-02-24, 22:18
Replies: 18
Views: 218
Posted By paulunderwood
n! the numerator is divisible by n, but each of...

n! the numerator is divisible by n, but each of the factors of k! and (n-k)! are not.

Here is a concrete example 7!/(5!*2!). Here 7 divides 7! but not 5! or 2!
Forum: Information & Answers 2021-02-24, 21:10
Replies: 18
Views: 218
Posted By paulunderwood
I should have been more explicit: each of the...

I should have been more explicit: each of the factors of k! and (n-k)! are less than n which is assumed to be prime. n would not divide anything smaller. For example 7 does not divide 5!
Forum: Information & Answers 2021-02-24, 20:49
Replies: 18
Views: 218
Posted By paulunderwood
Since for prime n we have a^n== a mod n then...

Since for prime n we have a^n== a mod n then a^(n-1) == 1. We can take a square root a^((n-1)/2) and expect the result to be +- 1. If it is 1 and we can take another square root by dividing the...
Forum: Information & Answers 2021-02-24, 20:36
Replies: 18
Views: 218
Posted By paulunderwood
Surely you did the binomial theorem. (a+b)^n =...

Surely you did the binomial theorem. (a+b)^n = a^n + binom(n,1)*a^(n-1)*b + binom(n,2)*a^(n-2)*b^2 + ... + b^n. Now think about a particular binom(n,k). It is n!/(k!*(n-k)!). If n was prime then...
Forum: Data 2021-02-24, 09:46
Replies: 549
Views: 73,534
Posted By paulunderwood
Thanks, Jean. I merely got the candidate...

Thanks, Jean.

I merely got the candidate from www.mersenne.ca. I might have run a 3-PRP to be sure-ish. Anyway, Primo does a quick Fermat+Lucas à la BPSW before embarking on a lengthy ECPP path.
Forum: Miscellaneous Math 2021-02-24, 09:13
Replies: 72
Views: 2,137
Posted By paulunderwood
Are you seriously thinking of spreading this...

Are you seriously thinking of spreading this calculation over a supercomputer running Mathematica? At the current wavefront you would need to calculate a polynomial with ~2^109000000 terms, let alone...
Forum: Data 2021-02-24, 06:42
Replies: 549
Views: 73,534
Posted By paulunderwood
@James, M82939 cofactor...

@James,

M82939 cofactor (https://primes.utm.edu/primes/page.php?id=132049) is certified prime
Forum: And now for something completely different 2021-02-24, 06:38
Replies: 47
Views: 14,086
Posted By paulunderwood
M82939 cofactor is certified...

M82939 cofactor is certified (https://primes.utm.edu/primes/page.php?id=132049)

FactorDB entry (http://www.factordb.com/index.php?id=1100000000213099295)
Forum: Software 2021-02-23, 01:37
Replies: 12
Views: 3,406
Posted By paulunderwood
Exit mprime and in that directory run sudo chown...

Exit mprime and in that directory run sudo chown storm5510:storm5510 *. Then start mprime as non-root and no sudo.
Forum: Miscellaneous Math 2021-02-22, 21:03
Replies: 10
Views: 309
Posted By paulunderwood
2^13-1== 8191 is prime. 2^8191-1 is not. Easy to...

2^13-1== 8191 is prime. 2^8191-1 is not. Easy to check.
Forum: Software 2021-02-22, 17:06
Replies: 7
Views: 222
Posted By paulunderwood
Run top and press k (and kill the pid with...

Run top and press k (and kill the pid with SIGTERM?),
Forum: Miscellaneous Math 2021-02-22, 01:52
Replies: 10
Views: 309
Posted By paulunderwood
https://primes.utm.edu/mersenne/index.html#unknown...

https://primes.utm.edu/mersenne/index.html#unknown

Proving a ~10^51217599719369681875006054625051616349 digit number prime is beyond all known technolgy.
Forum: Miscellaneous Math 2021-02-21, 18:57
Replies: 18
Views: 971
Posted By paulunderwood
Question Big question

Are the solutions to b^2 = 2^i*3^j + 1 finite? (i, j >= 0)

I have only found b in [2, 3, 5 ,7, 17].
Forum: Miscellaneous Math 2021-02-21, 16:17
Replies: 18
Views: 971
Posted By paulunderwood
Latest test: ...

Latest test:

{EulerPRP(a,n)=Mod(a,n)^((n-1)/2)==kronecker(a,n);}

{
tst(n,a)=
gcd(a+3,n)==1&&
gcd(3*a+1,n)==1&&
gcd(a^2+6*a+1,n)==1&&
kronecker(a,n)==-1&&
Forum: Miscellaneous Math 2021-02-21, 05:21
Replies: 18
Views: 971
Posted By paulunderwood
Question All Euler-PRP = -1

I am now testing:

{tst(n,a)=kronecker(a,n)==-1&&kronecker(a-1,n)==-1&&kronecker(a+1,n)==-1&&
Mod(a,n)^((n-1)/2)==-1&&Mod(a-1,n)^((n-1)/2)==-1&&Mod(a+1,n)^((n-1)/2)==-1&&...
Forum: Miscellaneous Math 2021-02-20, 18:28
Replies: 72
Views: 2,137
Posted By paulunderwood
There is a huge difference in run times between...

There is a huge difference in run times between checking Mp/f is integer and Mod(2,f)^p==1. Really. You should try it. Only a fool wouldn't.
Forum: Riesel Prime Search 2021-02-19, 19:42
Replies: 238
Views: 37,769
Posted By paulunderwood
That may be the smallest PRP. No proof of...

That may be the smallest PRP. No proof of primality, unless you have a new theorem :smile:
Forum: Miscellaneous Math 2021-02-19, 11:20
Replies: 72
Views: 2,137
Posted By paulunderwood
Okay grasshopper, now implement modular reduction...

Okay grasshopper, now implement modular reduction over Mp either in Pari/GP or MMA.

Then do some shallow trial division by 2*k*p+1.

I expect this to speed up your codes by 20 fold.
Forum: Miscellaneous Math 2021-02-19, 10:41
Replies: 18
Views: 971
Posted By paulunderwood
Arrow Rationale

The rationale for the above test:


Matrix: [a,a;1,a]
Characteristic eqn: y^2-2*a+a^2-a==0
Solutions: y = a +- sqrt(a)
"Strong" transform: z^2-2*(a+1)/(a-1)*z+1=0 and Euler-(a^2-a)==-1
...
Forum: Riesel Prime Search 2021-02-19, 08:30
Replies: 238
Views: 37,769
Posted By paulunderwood
Congrats to Serge Batalov for the smallest known...

Congrats to Serge Batalov for the smallest known 1 million digit prime:

10^999999 + 308267*10^292000 + 1 (https://primes.utm.edu/primes/page.php?id=131964)

Can this be limbo-ed?
Forum: Miscellaneous Math 2021-02-19, 06:29
Replies: 72
Views: 2,137
Posted By paulunderwood
You are still calculaing Mp for composite p. Read...

You are still calculaing Mp for composite p. Read Wiki on Mersenne primes. p needs to be prime. Put an if statement at the beginning of the loop to check p is prime.

This is bad: s = 4; Do[s =...
Forum: Miscellaneous Math 2021-02-18, 14:26
Replies: 18
Views: 971
Posted By paulunderwood
I found a counterexample to the above test for...

I found a counterexample to the above test for n=137975195359.

I am now testing:


{tst(n,a)=kronecker(a,n)==-1&&kronecker(a-1,n)==1&&kronecker(a+1,n)==-1&&...
Forum: Miscellaneous Math 2021-02-18, 02:33
Replies: 18
Views: 971
Posted By paulunderwood
Talking Unique solution test

{
tst(n,a)=local(A=a^2+a);
kronecker(A,n)==-1&&
kronecker(a,n)==-1&&
Mod(A,n)^((n-1)/2)==-1&&
Mod(a,n)^((n-1)/2)==-1&&
Mod(Mod(z,n),z^2-(4*a/(a-1)-2)*z+1)^((n+1)/2)==-1
}
Forum: GPU Computing 2021-02-17, 23:41
Replies: 358
Views: 50,620
Posted By paulunderwood
My Corsair PSUs have 8 or 10 year warranty :two...

My Corsair PSUs have 8 or 10 year warranty :two cents:
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