Carol / Kynea search (Near-power primes)

[lalera]

hi,
here are the results for near-cube numbers
done with cksieve v3.1.7 and openpfgw
b=74, n=1 to 10000
(74^27+1)^3-2
(74^2564-1)^3-2
(74^9291+1)^3-2

[lalera]

hi,
here are the results for near-cube numbers
done with cksieve v3.1.8 and openpfgw
b=6, n=1 to 10000
(6^1-1)^3+2
(6^3-1)^3+2
(6^2+1)^3-2
(6^3+1)^3-2
(6^44+1)^3+1*(-2)
(6^48+1)^3+1*(-2)
(6^57+1)^3+1*(-2)
(6^188+1)^3+1*(-2)
(6^624-1)^3-1*(-2)
(6^738+1)^3+1*(-2)
(6^1284-1)^3-1*(-2)
(6^1571-1)^3-1*(-2)
(6^2324-1)^3-1*(-2)
(6^2907-1)^3-1*(-2)
(6^5418-1)^3-1*(-2)
(6^6161-1)^3-1*(-2)
-
b=12, n=1 to 10000
(12^2-1)^3+2
(12^2+1)^3-2
(12^3+1)^3-2
(12^24-1)^3-1*(-2)
(12^122-1)^3-1*(-2)
-
b=74, n=1 to 10000
(74^27+1)^3+1*(-2)
(74^9291+1)^3+1*(-2)

[Batalov]

And (2^471043+1)^3-2 is prime!

[paulunderwood]

Originally Posted by Batalov

And (2^471043+1)^3-2 is prime!

Congrats

How about near-quartics? With a little factorisation, a CHG proof could be done

[Batalov]

No, that would be a lot of factorization!
Doing just one additional % (up to still unmanageable 26% CHG of this size) is getting > 3882 additional digits of factors - that's unrealistic.

Besides, there exist tons of near-4th-powers primes (x⁴+1) - they are Generalized Fermats.

For this near-cube, btw, to avoid K-P proof, we observe that p+1 has a factor of 5209 * which pushes the log(factored(N+1))/log(N) strictly over 1/3.
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* EDIT: ... and 8968913743