View Single Post
Old 2020-02-15, 06:14   #4
LaurV
Romulan Interpreter
 
LaurV's Avatar
 
Jun 2011
Thailand

100000101100102 Posts
Default

yeah, well.. the second form, which I rendered like below (probably a simpler form still exists for this one too) has no primes in sight, but there is no reason why a prime should not be there, the factoring looks like our crus lists, haha, and we would need a sieving program for it, because, with 19 dividing every third, 11 and 17 dividing every 8th, 10th, etc, the chances of finding a prime is very slim, but they should still come in an infinite amount, unless some aurifeuillian factorization unknown to us, or auriferous or platinodiamantine or whatever is called. By construction, these numbers are not divisible by 2, 3 (obviously, they are 1 (mod 12)), and 5, 7 (not so obvious, but see below) and that makes it harder to find a prime. But the prime is there, well hidden..

Code:
gp > ?doze41
doze41 =  (n)->49*12*(12^(2*n)-1)/143+1
gp > for(i=1,10,print(digits(doze41(i),12)))
[4, 1, 1]
[4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1]
gp > for(n=1,100,print(n": "factorint(doze41(n))~[1,]))
1: [19, 31]
2: [11, 23, 337]
3: [17, 722237]
4: [13, 19, 1979, 3617]
5: [571, 991, 449929]
6: [177823, 206171347]
7: [19, 277859424790111]
8: [37, 131, 13159, 105173, 113329]
9: [2837, 38587299124605977]
10: [19, 17189, 48268299300277811]
11: [17, 133530286335052494259517]
12: [367, 37560247, 23713556318104789]
13: [11, 19, 23, 35153, 1993859951, 139708400149]
14: [41, 21070110346361, 7846310875960621]
15: [1109, 880130606633990466566760020441]
16: [19, 31, 53, 21023, 78234938983, 2737502719302277]
17: [13, 37, 998861, 279663973627, 150631914918952787]
18: [101, 797, 10891, 1328911, 4006823, 624342125794134911]
19: [17, 19, 5527, 231898913, 2642684909, 383612326598340557]
20: [189767, 2958811, 96442165805129, 1116056700437206537]
21: [8273, 88379, 35125823, 338855864639826302126776175129]
22: [19, 823, 12541, 6390436308030046891653188795484059219533]
23: [3725522325449, 48438583726617962727111389675540810021]
24: [11, 23, 67, 12563393, 505124491, 9544230624930767, 25310427402129151]
25: [19, 83, 643, 1789, 139939, 161312028629, 91378718923313704966951677181]
26: [37, 463, 877, 2387291, 121589407, 123561358204325220849241943959288199]
27: [17, 4564357509721325038561737361681120186114742434887025935677]
28: [19, 1871, 100696837, 3667879961, 851006222623598233419959001107582844037]
29: [36955415772205820717, 1524981620206945422383, 28550311910196094306159]
30: [13, 14718871, 40827524265371788247316943, 29658224926068953911305277722409]
31: [19, 31, 71, 2339, 1469022047, 12220606596293613370899217920874878772972292555153]
32: [242276653, 415099039, 47772475059814551788252488123598816293657760084470783]
33: [50591, 55762433, 121017041968289563417, 2026478573464457245639635066584383583179]
  *** factorint: user interrupt after 15,460 ms
  ***   Break loop: <Return> to continue; 'break' to go back to GP prompt
break>

Last fiddled with by LaurV on 2020-02-15 at 06:28
LaurV is offline   Reply With Quote