## Sunday, April 05, 2020

### [pnioejoi] Some generalized Fermat primes

The following Pari/GP code prints out the indices (i) of the first 20 Generalized Fermat Primes i^2^x+1, for values of x between 0 and 10.

for(x=0 , 10 , print1(x," :") ; c=0 ; for(i=0 , +oo , p=i^2^x+1 ; if(ispseudoprime(p) , print1(" ",i) ; c++ ; if(20==c , print() ; break))))

0 : 1 2 4 6 10 12 16 18 22 28 30 36 40 42 46 52 58 60 66 70
1 : 1 2 4 6 10 14 16 20 24 26 36 40 54 56 66 74 84 90 94 110
2 : 1 2 4 6 16 20 24 28 34 46 48 54 56 74 80 82 88 90 106 118
3 : 1 2 4 118 132 140 152 208 240 242 288 290 306 378 392 426 434 442 508 510
4 : 1 2 44 74 76 94 156 158 176 188 198 248 288 306 318 330 348 370 382 396
5 : 1 30 54 96 112 114 132 156 332 342 360 376 428 430 432 448 562 588 726 738
6 : 1 102 162 274 300 412 562 592 728 1084 1094 1108 1120 1200 1558 1566 1630 1804 1876 2094
7 : 1 120 190 234 506 532 548 960 1738 1786 2884 3000 3420 3476 3658 4258 5788 6080 6562 6750
8 : 1 278 614 892 898 1348 1494 1574 1938 2116 2122 2278 2762 3434 4094 4204 4728 5712 5744 6066
9 : 1 46 1036 1318 1342 2472 2926 3154 3878 4386 4464 4474 4482 4616 4688 5374 5698 5716 5770 6268
10 : 1 824 1476 1632 2462 2484 2520 3064 3402 3820 4026 6640 7026 7158 9070 12202 12548 12994 13042 15358

For example, the entry "30" in row 5 means that 30^2^5+1 = 30^32+1 is prime.

Are these indices unusually smooth (in the context of integer factorization)?  (Other than the fact that they are always even except for 1.)  We do see numbers like 30, 360, and 1200.

There's a prefix in the first row 1 2 4 6 10 12 that shortens by exactly one each successive row.  Is this triangular pattern just a coincidence?

The largest prime on the table, 15358^1024+1 has size 14241 bits.

The matrix has been extended in both directions by others.

The second column (2 2 2 2 2 30 102 120 278 46 824...) is OEIS A056993.

Currently, the first difficult row is row 21.