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Forum: Miscellaneous Math 2021-03-01, 20:48
Replies: 1
Views: 35
Posted By paulunderwood
List of candidates by crazy formula

Based on the "test" lift(Mod(420,p^2-1)^p)%p==0 I get this partial list of exponents


3
5
7
11
13
17
19
Forum: Software 2021-02-28, 15:28
Replies: 7
Views: 323
Posted By paulunderwood
If you have tried llrCuda out please post how...

If you have tried llrCuda out please post how well it does. How many AVX2 cores is it equivalent to on a top end nVidia card?
Forum: Hardware 2021-02-28, 08:23
Replies: 162
Views: 15,824
Posted By paulunderwood
Can you run mprime on it out of the box or do you...

Can you run mprime on it out of the box or do you have to compile it to get it working on your Phi?
Forum: Hardware 2021-02-26, 13:13
Replies: 162
Views: 15,824
Posted By paulunderwood
There are a number of 56/60 22nm Xeon Phi on...

There are a number of 56/60 22nm Xeon Phi on ebay.co.uk at the moment for about £100. What are the adavantages? -- 60 GHz is lot; Do the have AVX2?. Apart from being 250w what are the...
Forum: Information & Answers 2021-02-24, 22:52
Replies: 20
Views: 444
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: 20
Views: 444
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: 20
Views: 444
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: 20
Views: 444
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: 20
Views: 444
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: 74,226
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: 2,224
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: 74,226
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: 14,341
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: 3,515
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: 323
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: 8
Views: 262
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: 323
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: 1,020
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: 1,020
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: 1,020
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: 2,224
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: 38,189
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: 2,224
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: 1,020
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: 38,189
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?
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