mersenneforum.org Carol / Kynea half squares
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2020-03-29, 22:57   #1
sweety439

Nov 2016

215910 Posts
Carol / Kynea half squares

Quote:
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.

2020-03-29, 22:59   #2
sweety439

Nov 2016

1000011011112 Posts

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
Attached Files
 Carol for b le 512.txt (9.9 KB, 26 views) Kynea for b le 512.txt (10.1 KB, 14 views)

 2020-03-29, 23:01 #3 sweety439     Nov 2016 41578 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 23·5·227 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.)

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