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2020-10-23, 11:16   #12
mart_r

Dec 2008
you know...around...

60810 Posts

Quote:
 Originally Posted by Dr Sardonicus Once upon a time, long long ago, I posted a link to a paper discussing this, Prime Number Races.
It's like you can read my mind, I have a printout of that paper (actually one of the first math papers I have made a printout of) and wanted to look into it today to recap on logarithmic measures.

2020-10-23, 12:37   #13
Dr Sardonicus

Feb 2017
Nowhere

2×1,901 Posts
Sequence of 3159 numbers containing 447 primes

Quote:
 Originally Posted by R2357 Anyway, if there indeed is such a sequence, then the first or the occurrence will have been reached by 32 589 158 477 190 044 730, thus way below the lower band of 10^174.
By 53#? I don't know where in the world you got that idea.

2020-10-23, 14:11   #14
R2357

"Ruben"
Oct 2020
Nederland

2×19 Posts

Quote:
 Originally Posted by Dr Sardonicus By 53#? I don't know where in the world you got that idea.
Simply because in any sequence 53#*n to 53#*n+1, all the numbers that are congruent to x*{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# will be divisible by a prime up to 53.

59²=3481, 3481>3159 and 59 is the next prime after 53.

That's why I think the conjecture likely!

 2020-10-23, 16:12 #15 mart_r     Dec 2008 you know...around... 25·19 Posts Just because I can... Take, for instance, n = 1566280308578217520031412816790827048467516641360946961779273951 and set p = 163#, or in full, p = 5766152219975951659023630035336134306565384015606066319856068810 then the numbers n+x are all coprime to p for the following 447 values of x: Code: 0 2 6 8 12 20 30 32 42 48 50 56 68 72 78 90 92 96 98 102 110 116 128 132 138 146 152 156 158 162 176 186 200 210 212 216 228 230 240 242 246 252 260 266 272 278 282 288 300 306 308 312 320 326 336 338 342 350 366 372 378 380 386 396 398 420 422 428 432 438 440 450 462 470 476 482 488 492 498 506 510 512 516 518 536 546 548 558 572 576 590 600 602 606 618 620 638 648 650 656 662 672 690 692 702 708 716 722 726 740 746 756 762 768 776 782 792 798 806 810 812 818 828 840 846 858 860 866 870 872 876 882 888 890 900 912 926 930 938 942 950 956 960 966 968 980 986 992 1008 1020 1022 1026 1040 1052 1056 1058 1062 1068 1080 1082 1086 1098 1110 1112 1118 1122 1142 1146 1148 1166 1170 1176 1178 1188 1190 1196 1202 1212 1220 1236 1238 1250 1260 1262 1266 1272 1278 1290 1296 1302 1308 1328 1332 1346 1350 1352 1356 1370 1376 1392 1400 1406 1416 1418 1428 1430 1436 1442 1448 1458 1470 1476 1478 1488 1496 1500 1502 1520 1530 1532 1538 1542 1560 1566 1568 1580 1590 1602 1608 1610 1626 1632 1638 1640 1646 1652 1656 1670 1680 1682 1692 1706 1712 1716 1722 1728 1730 1748 1758 1766 1770 1772 1790 1796 1800 1806 1812 1818 1826 1832 1836 1838 1848 1850 1862 1866 1868 1878 1880 1892 1896 1898 1902 1910 1916 1926 1932 1938 1940 1946 1962 1980 1982 1988 1992 2000 2006 2016 2022 2028 2036 2046 2048 2052 2060 2066 2070 2072 2076 2078 2088 2102 2108 2112 2120 2130 2132 2148 2162 2178 2186 2190 2192 2198 2202 2210 2226 2240 2252 2256 2258 2268 2280 2286 2310 2312 2316 2318 2322 2336 2342 2352 2358 2360 2366 2372 2388 2396 2400 2406 2408 2412 2430 2438 2442 2450 2456 2462 2468 2478 2486 2490 2492 2496 2508 2510 2522 2528 2538 2540 2552 2562 2570 2576 2580 2582 2592 2598 2606 2616 2618 2622 2630 2640 2648 2652 2658 2660 2666 2676 2682 2690 2696 2706 2708 2718 2732 2736 2742 2750 2756 2760 2762 2778 2786 2790 2798 2808 2820 2822 2826 2828 2850 2856 2858 2862 2868 2882 2888 2892 2900 2910 2912 2918 2928 2930 2940 2952 2958 2966 2970 2972 2982 2990 2996 3002 3018 3036 3042 3050 3056 3060 3066 3072 3080 3086 3092 3098 3102 3110 3122 3126 3128 3138 3150 3152 3158 And, viola, you have 447 potential prime number candidates in an interval of 3159. The only thing you have to do now is to add p to n as many times until you find an example where all 447 numbers are prime. Simple as that
 2020-10-23, 18:12 #16 R2357   "Ruben" Oct 2020 Nederland 2616 Posts Co prime doesn't mean prime, and I still don't understand why the odds are to think that the conjecture is wrong, okay, it's incompatible with the first, but there is no particular reason to think it's false, why would we rather think that.
 2020-10-23, 18:38 #17 mart_r     Dec 2008 you know...around... 25·19 Posts Then, I'm afraid, the odds that anyone can help you understand are about as good as the odds of finding an actual prime-447-tuplet. Edit, FWIW: If the first Hardy-Littlewood conjecture was wrong, this would have far-reaching consequences for prime number theory, probably of a magnitude as disproving the Riemann hypothesis. The second conjecture is based merely on human intuition; conjectures of this kind have been proven wrong a myriad of times in the past. Last fiddled with by mart_r on 2020-10-23 at 19:13
2020-10-23, 18:46   #18
Dr Sardonicus

Feb 2017
Nowhere

2×1,901 Posts

Quote:
 Originally Posted by mart_r Just because I can... Take, for instance, n = 1566280308578217520031412816790827048467516641360946961779273951 and set p = 163#, or in full, p = 5766152219975951659023630035336134306565384015606066319856068810 then the numbers n+x are all coprime to p for the following 447 values of x: Code: 0 2 6 8 12 3158 And, viola, you have 447 potential prime number candidates in an interval of 3159. The only thing you have to do now is to add p to n as many times until you find an example where all 447 numbers are prime. Simple as that
There is an "admissibility condition" to the effect that there can be no inevitable prime factor.

To check this, I told Pari-GP to compute the degree-447 polynomial

f = x*(x+2)*(x+6)*(x+8)*(x+12)*...*(x+3158)

and then to compute Mod(Mod(1,p)*f, x^p - x) for all primes p up to 447, to see if any were zero. None were. (Of course, for p > 447, the degree of f is less that p, so the polmod will be Mod(1,p)*f, which isn't 0.)

Looks like you're good to go.

Of course, as indicated by 447 tuples calculations, finding an example where the numbers are all prime might take a while.

Last fiddled with by Dr Sardonicus on 2020-10-24 at 00:31 Reason: awkward phrasing; nixfig ostpy

2020-10-23, 19:20   #19
R2357

"Ruben"
Oct 2020
Nederland

1001102 Posts

Quote:
 Originally Posted by R2357 Simply because in any sequence 53#*n to 53#*n+1, all the numbers that are congruent to x*{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# will be divisible by a prime up to 53. 59²=3481, 3481>3159 and 59 is the next prime after 53. That's why I think the conjecture likely!
Okay, in my calculation, I overlooked the fact that numbers congruent to a composite up to 53# divisible by primes from 59 onwards, modulo 53# may be prime, mea culpa :|

But still, there remains two conditions :
- first the positions of the 3159 numbers must contain 447 numbers that are congruent to numbers not divisible by {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53#;
- secondly, the sequence chosen must not contain more numbers divisible by primes from 59 onwards than the difference (positive of course) between the numbers congruent to numbers not divisible by {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# and 447.

Good luck successfully reaching the two above conditions :)

2020-10-24, 14:27   #20
sweety439

Nov 2016

9C016 Posts

Quote:
 Originally Posted by R2357 Hello, I've looked at the way primes behave in sequences which have a primorial length and, as I mentionned a few weeks ago, I came across the 2nd Hardy-Littlewood conjecture, which was expained to me as being incompatible with the first Hardy-Littlewood conjecture. What I don't understand is why do we believe that it's the first one which is right, personnaly, I believe it's the second, here's why I think so : The second Hardy-Littlewood conjecture states that with x>1, pi(x)>=pi(x+y)-pi(y). That seems to me as very probable! Already, the first sequence of 30 contains 10 primes, none of the others will contain more than 7. Why would this conjecture seem to be false?
If Schinzel's hypothesis H is true, then the second Hardy-Littlewood conjecture is false, since Schinzel's hypothesis H covers both Bunyakovsky conjecture and Dickson's conjecture, Dickson's conjecture covers first Hardy–Littlewood conjecture, but the first & second Hardy–Littlewood conjecture cannot be both true since ....

primepi(3159) = 446, but there may be 3159 consecutive positive integers which contain 447 primes (unlike the case of 6 consecutive positive integers, any 6 consecutive positive integers >= {4,5,6,7,8,9} contain at most 2 primes, where prmiepi(6) = 3), this is a case of law of small numbers.

However, Schinzel's hypothesis H is widely believed to be true, like generalized Riemann hypothesis and abc conjecture.

Last fiddled with by sweety439 on 2020-10-24 at 14:28

2020-10-26, 17:07   #21
R2357

"Ruben"
Oct 2020
Nederland

2×19 Posts

Quote:
 Originally Posted by mart_r The second conjecture is based merely on human intuition;
I don't understand what you really mean by "human intuition", when we look at the primes from 1 to 3159, this makes more than 14% of the natural numbers, the non-multiples of one and two digit numbers make up barely 12% of them.

furthermore, it's not at all similar to the twin-prime conjecture, because here, we're looking for a bigger prime density than in the beginning.

And it's not just an intuition : each p# takes away the number of possible primes in pn-1# for every p#, not to mention all of the non-primes divisible by numbers bigger than p : in 17#, in the first sequence of 17# is already made out of more composites only by numbers from19, than primes, I let you imagine in for example, let's say 10³³*17# to 10³³+1*17# :)

2020-10-26, 19:30   #22
CRGreathouse

Aug 2006

10111001100102 Posts

Quote:
 Originally Posted by R2357 I don't understand what you really mean by "human intuition", when we look at the primes from 1 to 3159, this makes more than 14% of the natural numbers, the non-multiples of one and two digit numbers make up barely 12% of them.
There are 446 primes up to 3159, so 14.1%. There are 422 numbers coprime to 1, 2, ..., 99, or 13.4%. What does this show? 3159 isn't a counterexample to the second Hardy-Littlewood conjecture. But there are infinitely many other cases to check, and we should be cautious in generalizing from 'one case holds' to 'all cases hold': as is often the case in number theory, the first counterexample may be large.

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