Carol / Kynea half squares

sweety439 · 2020-03-29, 22:57

Quote:

Originally Posted by rogue

I've decided to start a new thread for a coordinated Carol / Kynea search. Carol and Kynea primes are subset of what is known as Near Square Primes. These primes have the form of (b^n-1)^2-2 (Carol) and (b^n+1)^2-2 (Kynea). Steven Harvey has coordinated the search in the past and still coordinates the base 2 search here. This thread is for those who want to search other bases. Although his page has some other bases on it, the page is incomplete and it is unknown to me if there are any gaps or if bugs might have causes previous searches to miss some primes.

I am already working on base 2 in another thread of the sub-forum and am coordinating that effort with Steven directly.

Odd bases can be skipped because the Carol/Kynea number is always even. I could modify the sieve to test (b^n+/-2)^2+/-1 for odd b, but that is for a different project somewhere down the road.

pfgw (for base 2) is about 15% faster than llr, so pfgw is recommended for testing this form at this time.
This form is supported by PRPNet, so you can find that elsewhere if you want to use it to distribute PRP testing.

Please post primes in this thread.

Please go here to see a list of current reservations and to find the current version of cksieve, but continue to use this thread to submit and complete your reservations.

Attached are sieving files available for testing. You may want to verify that they have been sufficiently sieved before beginning testing.

For odd b, I suggest ((b^n-1)^2-2)/2 (Carol) and ((b^n+1)^2-2)/2 (Kynea), they are half Carol/Kynea, like the generalized half Fermat.

sweety439 · 2020-03-29, 22:59

Quote:

Originally Posted by sweety439

For odd b, I suggest ((b^n-1)^2-2)/2 (Carol) and ((b^n+1)^2-2)/2 (Kynea), they are half Carol/Kynea, like the generalized half Fermat.

These are status for both sides for b<=512, searched up to n=1024

sweety439 · 2020-03-29, 23:01

Can someone find a prime of the form ((81^n-1)^2-2)/2, ((215^n-1)^2-2)/2, ((319^n-1)^2-2)/2, ((73^n+1)^2-2)/2, ((109^n+1)^2-2)/2, ((205^n+1)^2-2)/2, etc?

Batalov · 2020-03-29, 23:41

It is very easy.

((205^3651+1)^2-2)/2
((205^4133+1)^2-2)/2
((205^4620+1)^2-2)/2
((215^12694-1)^2-2)/2
((319^11276-1)^2-2)/2
((319^5513-1)^2-2)/2
((73^1275+1)^2-2)/2
((73^2004+1)^2-2)/2
etc

However, the problem with these half-near-squares is that you will not be able to prove (most of) them. (Take these PRP-1 and PRP+1: there is nothing immediately smooth about them.)

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2020-03-29, 23:01	#3
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×3×613 Posts	Can someone find a prime of the form ((81^n-1)^2-2)/2, ((215^n-1)^2-2)/2, ((319^n-1)^2-2)/2, ((73^n+1)^2-2)/2, ((109^n+1)^2-2)/2, ((205^n+1)^2-2)/2, etc?

2020-03-29, 23:41	#4
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 10021₁₀ Posts	It is very easy. ((205^3651+1)^2-2)/2 ((205^4133+1)^2-2)/2 ((205^4620+1)^2-2)/2 ((215^12694-1)^2-2)/2 ((319^11276-1)^2-2)/2 ((319^5513-1)^2-2)/2 ((73^1275+1)^2-2)/2 ((73^2004+1)^2-2)/2 etc However, the problem with these half-near-squares is that you will not be able to prove (most of) them. (Take these PRP-1 and PRP+1: there is nothing immediately smooth about them.)