Martin Hopf's prime numbers

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2:57 PM

$f(1 \ 000)+1$ works quite well with PFGW:

8273631704085440....0000000000000001 is composite: RES64: [BEDCCADA556094C9] (221.6781s+0.0279s).

For $f(n) \pm 1$ with $n \le 1 \ 000$ there are no more PRP's to be announced.

Peter

Wow, both types ? And upto $n=1\ 000$ ? Impressing !

Martin Hopf

Pre-sieve small factors with Pari/Gp and then 3-PRP test with PFGW. That's all.

Peter

Good job however. Let us invent another expression, hopefully not already done by someone !

Martin Hopf

Yeah! An expression with lots and lots primes please.

Peter

What abouth this ?

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Your results
status (?) n digits number
P 468 195 I(468)^2+I(468)+1<195> = 8193345517...73<195>
499 numbers not shown. Reasons: 435 times incompletely factored 32 times composite 32 times fully factored
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Your results
status (?) n digits number
PRP * 958 400 I(958)^2+I(958)+1<400> = 5264397989...21<400>
PRP * 1352 565 I(1352)^2+I(1352)+1<565> = 2532848604...51<565>
498 numbers not shown. Reasons: 447 times incompletely factored 10 times fully factored 41 times composite
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Martin Hopf

3:13 PM

Peter

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Your results
status (?) n digits number
PRP * 1518 634 I(1518)^2+I(1518)+1<634> = 6130639719...41<634>
PRP * 1556 650 I(1556)^2+I(1556)+1<650> = 4683456098...07<650>
PRP * 1563 653 I(1563)^2+I(1563)+1<653> = 3948147935...07<653>
PRP * 1861 778 I(1861)^2+I(1861)+1<778> = 1422417235...83<778>
496 numbers not shown. Reasons: 453 times incompletely factored 40 times composite 3 times fully factored
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appears to be a good generator !

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Your results
status (?) n digits number
PRP * 2788 1165 I(2788)^2+I(2788)+1<1165> = 4131540644...93<1165>
499 numbers not shown. Reasons: 450 times incompletely factored 47 times composite 2 times fully factored
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Now you have something you can compare with the PFGW-output. This time it is straight forward to search for the primes. Shall I do that or do you want to do that ?

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Your results
status (?) n digits number
PRP * 3238 1353 I(3238)^2+I(3238)+1<1353> = 5069769969...81<1353>
499 numbers not shown. Reasons: 453 times incompletely factored 44 times composite 2 times fully factored
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Composites without known factor :

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Your results
status (?) n digits number
C 229 96 I(229)^2+I(229)+1<96> = 1040804665...71<96>
C 238 99 I(238)^2+I(238)+1<99> = 6013769174...81<99>
C 286 119 I(286)^2+I(286)+1<119> = 6949606563...53<119>
C 288 120 I(288)^2+I(288)+1<120> = 4763331200...93<120>
C 320 134 I(320)^2+I(320)+1<134> = 1130107496...91<134>
C 331 138 I(331)^2+I(331)+1<138> = 4475564714...31<138>
C 352 147 I(352)^2+I(352)+1<147> = 2681197042...01<147>
C 356 149 I(356)^2+I(356)+1<149> = 1259591883...07<149>

@MartinHopf

Martin Hopf

This is an easy syntax for PFGW

Peter

Exactly !

Martin Hopf

3:35 PM

for(n=1,10000,q=fibonacci(n)^2+fibonacci(n)+1;if(ispseudoprime(q),print1(n,", ")));
1, 2, 3, 4, 5, 6, 8, 11, 16, 24, 36, 43, 56, 64, 72, 158, 190, 213, 326, 416, 468, 958, 1352, 1518, 1556, 1563, 1861, 2788, 3238, 5787, 7078

Peter

3:50 PM

FF 96 (show) (I(236)^2+I(236)+1)/721<96> = 916963804462443311227<21> · 5095161881213559243043<22> · 11964354360846459786302227<26> · 2177015302006330090219857661<28>

Martin Hopf

4:11 PM

no more PRP's for $n \le 20 \ 000$.

Peter

I think, you should continue with PFGW , although PARI/GP is quite fast as well upto this limit.

Martin Hopf

Of course. PFGW:

F(7078)^2+F(7078)+1 is 3-PRP! (0.1845s+0.0003s)

that was the last response we are now at $n=24603$.

WOW:

F(24837)^2+F(24837)+1 is 3-PRP! (2.5257s+0.0004s)

Peter