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 2005-07-25, 16:46 #12 alpertron     Aug 2002 Buenos Aires, Argentina 22×3×53 Posts By the way, I wrote a program in UBASIC to compute the lowest value of Q for which both n#+Q and n#-Q are Fermat pseudoprimes base 3: Code:  n Q 11 23 13 17 17 59 19 23 23 79 29 101 31 83 37 239 41 71 43 149 47 367 53 73 59 911 61 313 67 373 71 523 73 313 79 331 83 197 89 101 97 1493 101 523 103 293 107 577 109 2699 113 1481 127 1453 131 5647 137 647 139 419 149 757 151 4253 157 509 163 239 167 10499 173 191 179 4013 181 2659 191 617 193 6733 197 1297 199 971 211 10093 223 313 227 1871 229 883 233 443 239 1009 241 1471 251 12689 257 2861 263 8231 269 7127 271 1217 277 5471 281 7673 283 773 293 10987 307 3643 311 331 313 3533 317 8111 331 6427 337 2441 347 1627 349 9829 353 937 359 4973 367 373 373 10501 379 1327 383 2711 389 16481 397 14717 401 8093 409 26777 419 31271 421 677 431 31957 433 2503 439 42829 443 5519 449 57089 457 6491 461 1021 463 27919 467 941 479 22441 487 54091 491 10987 499 691 503 17669 509 27527 521 20731 523 4001 541 18859 547 3457 557 13337 563 75391 569 25237 571 29077 577 11593 587 11483 593 10691 599 11777 601 21149 607 162527 613 66403 617 31991 619 60889 631 5179 Notice that in no case is Q >= n2, so all these values of Q are prime numbers.
2005-07-25, 18:10   #13
AntonVrba

Jun 2005

2·72 Posts

Quote:
 Originally Posted by alpertron By the way, I wrote a program in UBASIC to compute the lowest value of Q for which both n#+Q and n#-Q are Fermat pseudoprimes base 3: Code:  n Q 11 23 13 17 17 59 etc Notice that in no case is Q >= n2, so all these values of Q are prime numbers.
Yes indeed they are all prime provable by other software (in my case mathematica or PFGW), demonstrating Conjecture 5 as True.

If Conjecture 5 can be prooven then a new algorithm for prooving numbers prime can be developped that will shorten the computing time

Last fiddled with by AntonVrba on 2005-07-25 at 18:17

 2005-07-25, 20:35 #14 flava     Feb 2003 11101102 Posts I don't know if it's of any interest, but here are some quick results for lowest Q for p(i)#+-Q both prime. All Q here are prime. Code: p(10)# + 101 p(10)# - 101 p(11)# + 83 p(11)# - 83 p(12)# + 239 p(12)# - 239 p(13)# + 71 p(13)# - 71 p(14)# + 149 p(14)# - 149 p(15)# + 367 p(15)# - 367 p(16)# + 73 p(16)# - 73 p(17)# + 911 p(17)# - 911 p(18)# + 313 p(18)# - 313 p(19)# + 373 p(19)# - 373 p(20)# + 523 p(20)# - 523 p(21)# + 313 p(21)# - 313 p(22)# + 331 p(22)# - 331 p(23)# + 197 p(23)# - 197 p(24)# + 101 p(24)# - 101 p(25)# + 1493 p(25)# - 1493 p(26)# + 523 p(26)# - 523 p(27)# + 293 p(27)# - 293 p(28)# + 577 p(28)# - 577 p(29)# + 2699 p(29)# - 2699 p(30)# +1481 p(30)# -1481 p(31)# + 1453 p(31)# - 1453 p(32)# + 5647 p(32)# - 5647 p(33)# + 647 p(33)# - 647 p(34)# + 419 p(34)# - 419 p(35)# + 757 p(35)# - 757 p(36)# + 4253 p(36)# - 4253 p(37)# + 509 p(37)# - 509 p(38)# + 239 p(38)# - 239 p(39)# + 10499 p(39)# - 10499 p(40)# + 191 p(40)# - 191 p(41)# + 4013 p(41)# - 4013 p(42)# + 2659 p(42)# - 2659 p(43)# + 617 p(43)# - 617 p(44)# + 6733 p(44)# - 6733 p(45)# + 1297 p(45)# - 1297 p(46)# + 971 p(46)# - 971 p(47)# + 10093 p(47)# - 10093 p(48)# + 313 p(48)# - 313 p(49)# + 1871 p(49)# - 1871 p(50)#+883 p(50)#-883 p(100)# + 18859 p(100)# - 18859 p(101)# + 3457 p(101)# - 3457 p(200)# + 99259 p(200)# - 99259 p(201)# + 90677 p(201)# - 90677
 2005-07-25, 22:07 #15 alpertron     Aug 2002 Buenos Aires, Argentina 22·3·53 Posts To satisfy my curiosity I extended the table up to 1000: Code:  n Q ln(Q)/ln(ln(n#)) 5 7 1.590 7 13 1.530 11 23 1.532 13 17 1.214 17 59 1.583 19 23 1.129 23 79 1.478 29 101 1.480 31 83 1.356 37 239 1.616 41 71 1.215 43 149 1.385 47 367 1.591 53 73 1.128 59 911 1.751 61 313 1.446 67 373 1.463 71 523 1.519 73 313 1.372 79 331 1.365 83 197 1.225 89 101 1.056 97 1493 1.651 101 523 1.397 103 293 1.253 107 577 1.388 109 2699 1.707 113 1481 1.562 127 1453 1.543 131 5647 1.815 137 647 1.348 139 419 1.247 149 757 1.358 151 4253 1.699 157 509 1.258 163 239 1.098 167 10499 1.843 173 191 1.039 179 4013 1.630 181 2659 1.540 191 617 1.247 193 6733 1.701 197 1297 1.376 199 971 1.313 211 10093 1.750 223 313 1.085 227 1871 1.416 229 883 1.268 233 443 1.134 239 1009 1.281 241 1471 1.345 251 12689 1.735 257 2861 1.455 263 8231 1.641 269 7127 1.608 271 1217 1.283 277 5471 1.548 281 7673 1.602 283 773 1.187 293 10987 1.654 307 3643 1.453 311 331 1.024 313 3533 1.437 317 8111 1.578 331 6427 1.532 337 2441 1.358 347 1627 1.284 349 9829 1.591 353 937 1.180 359 4973 1.464 367 373 1.015 373 10501 1.583 379 1327 1.226 383 2711 1.344 389 16481 1.646 397 14717 1.622 401 8093 1.517 409 26777 1.714 419 31271 1.736 421 677 1.090 431 31957 1.731 433 2503 1.302 439 42829 1.771 443 5519 1.427 449 57089 1.810 457 6491 1.447 461 1021 1.139 463 27919 1.680 467 941 1.121 479 22441 1.636 487 54091 1.776 491 10987 1.513 499 691 1.061 503 17669 1.584 509 27527 1.652 521 20731 1.603 523 4001 1.335 541 18859 1.581 547 3457 1.306 557 13337 1.520 563 75391 1.793 569 25237 1.615 571 29077 1.635 577 11593 1.486 587 11483 1.481 593 10691 1.467 599 11777 1.480 601 21149 1.570 607 162527 1.888 613 66403 1.744 617 31991 1.626 619 60889 1.724 631 5179 1.336 641 67411 1.735 643 6329 1.363 647 7907 1.396 653 12437 1.464 659 6803 1.368 661 2423 1.206 673 21269 1.540 677 92503 1.764 683 11321 1.438 691 117571 1.796 701 57649 1.683 709 124781 1.799 719 29833 1.578 727 53093 1.663 733 17851 1.495 739 149159 1.816 743 12157 1.432 751 46441 1.634 757 3919 1.256 761 51563 1.645 769 3677 1.243 773 35117 1.582 787 86981 1.717 797 3607 1.235 809 25847 1.530 811 78511 1.695 821 18301 1.474 823 23509 1.510 827 37663 1.578 829 97813 1.719 839 90697 1.706 853 56393 1.633 857 109253 1.729 859 32371 1.546 863 26347 1.514 877 67217 1.651 881 35521 1.554 883 113843 1.725 887 47581 1.594 907 36161 1.551 911 3407 1.201 919 35257 1.544 929 16417 1.430 937 25577 1.493 941 11933 1.380 947 140159 1.740 953 26203 1.492 967 170921 1.765 971 27827 1.498 977 12157 1.375 983 252727 1.816 991 2011 1.109 997 37813 1.536 Notice that the exponent is never greater than 2, so all numbers Q in this list are prime.
2005-07-26, 00:27   #16
alpertron

Aug 2002
Buenos Aires, Argentina

22·3·53 Posts

Quote:
 Originally Posted by R.D. Silverman Here's what makes me think the conjecture is probably false. Put N = p#. The probability that N+Q is prime is about 1/log(N+Q) ~ 1/p The probability that N-Q is also prime is ~1/p. We want to search over values of Q, so that both are prime. If we let Q go from 1 to K, then the probability of finding both prime for some Q is sum from Q = 1 to k of 1/p^2 and this is just k/p^2 which is small. for k ~ log^2 N. I expect that for some p's we will have to take k to be bigger than log^2 N, i.e. Q will be p1*p2 for p1,p2 > p. The problem is that 1/p^2 is quite small for large p.
Bob, please correct if I'm wrong.

p# is not a random number. It is multiple of all numbers from 1 to p.

So if gcd(Q, p#) = 1, it is true that gcd(p# + Q, p#) = 1.

Since p# + Q does not have any divisor less than or equal to p, the probability that p# + Q is prime should be much greater than T + Q when T is a random number near p#. The same argument can be said for p# - Q.

2005-07-26, 04:53   #17
AntonVrba

Jun 2005

2×72 Posts

First of all, fast and great work in producing that table up to 997#

Quote:
 Originally Posted by alpertron Notice that the exponent is never greater than 2, so all numbers Q in this list are prime.. .
On what basis is above statement made?

Quote:
 Originally Posted by alpertron Bob, please correct if I'm wrong. So if gcd(Q, p#) = 1, it is true that gcd(p# + Q, p#) = 1. .
You are not wrong - just logic. You can also say

If gcd(Q,p#)=x then gcd(p# + Q, p#) = x.

Last fiddled with by AntonVrba on 2005-07-26 at 04:59

2005-07-26, 10:18   #18
AntonVrba

Jun 2005

6216 Posts

I found following:
Quote:
 http://mathworld.wolfram.com/EuclidsTheorems.html Guy (1981, 1988) points out that while p1 p2…pn+1 is not necessarily prime, letting q be the next prime after p1 p2…pn +1, the number q – p1 p2…pn+1 is almost always a prime, although it has not been proven that this must always be the case.
Ignoring the obvious error of +1 it is much the same as I have conjectured here.

Last fiddled with by AntonVrba on 2005-07-26 at 10:22

 2005-07-26, 12:05 #19 alpertron     Aug 2002 Buenos Aires, Argentina 27348 Posts Since both p# + Q and p# - Q are prime, Q has no divisor less than or equal to p. So if p2 < Q => ln Q / ln p < 2 the number Q must be prime (this is the basis for the trial division method). My error was to compute ln(Q)/ln(ln(p#)). The correct numbers are even lower than the numbers I computed yesterday: Code:  p Q ln(Q)/ln(p) 5 7 1.209 7 13 1.318 11 23 1.308 13 17 1.105 17 59 1.439 19 23 1.065 23 79 1.394 29 101 1.371 31 83 1.287 37 239 1.517 41 71 1.148 43 149 1.330 47 367 1.534 53 73 1.081 59 911 1.671 61 313 1.398 67 373 1.408 71 523 1.468 73 313 1.339 79 331 1.328 83 197 1.196 89 101 1.028 97 1493 1.598 101 523 1.356 103 293 1.226 107 577 1.361 109 2699 1.684 113 1481 1.544 127 1453 1.503 131 5647 1.772 137 647 1.316 139 419 1.224 149 757 1.325 151 4253 1.665 157 509 1.233 163 239 1.075 167 10499 1.809 173 191 1.019 179 4013 1.600 181 2659 1.517 191 617 1.223 193 6733 1.675 197 1297 1.357 199 971 1.299 211 10093 1.723 223 313 1.063 227 1871 1.389 229 883 1.248 233 443 1.118 239 1009 1.263 241 1471 1.330 251 12689 1.710 257 2861 1.434 263 8231 1.618 269 7127 1.586 271 1217 1.268 277 5471 1.530 281 7673 1.587 283 773 1.178 293 10987 1.638 307 3643 1.432 311 331 1.011 313 3533 1.422 317 8111 1.563 331 6427 1.511 337 2441 1.340 347 1627 1.264 349 9829 1.570 353 937 1.166 359 4973 1.447 367 373 1.003 373 10501 1.564 379 1327 1.211 383 2711 1.329 389 16481 1.628 397 14717 1.604 401 8093 1.501 409 26777 1.695 419 31271 1.714 421 677 1.079 431 31957 1.710 433 2503 1.289 439 42829 1.753 443 5519 1.414 449 57089 1.793 457 6491 1.433 461 1021 1.130 463 27919 1.668 467 941 1.114 479 22441 1.623 487 54091 1.761 491 10987 1.502 499 691 1.052 503 17669 1.572 509 27527 1.640 521 20731 1.589 523 4001 1.325 541 18859 1.564 547 3457 1.292 557 13337 1.502 563 75391 1.773 569 25237 1.598 571 29077 1.619 577 11593 1.472 587 11483 1.466 593 10691 1.453 599 11777 1.466 601 21149 1.556 607 162527 1.872 613 66403 1.730 617 31991 1.615 619 60889 1.714 631 5179 1.327 641 67411 1.720 643 6329 1.354 647 7907 1.387 653 12437 1.455 659 6803 1.360 661 2423 1.200 673 21269 1.530 677 92503 1.754 683 11321 1.430 691 117571 1.786 701 57649 1.673 709 124781 1.788 719 29833 1.566 727 53093 1.651 733 17851 1.484 739 149159 1.804 743 12157 1.423 751 46441 1.623 757 3919 1.248 761 51563 1.635 769 3677 1.235 773 35117 1.574 787 86981 1.706 797 3607 1.226 809 25847 1.517 811 78511 1.683 821 18301 1.463 823 23509 1.499 827 37663 1.568 829 97813 1.710 839 90697 1.696 853 56393 1.621 857 109253 1.718 859 32371 1.537 863 26347 1.506 877 67217 1.640 881 35521 1.545 883 113843 1.716 887 47581 1.587 907 36161 1.541 911 3407 1.194 919 35257 1.535 929 16417 1.420 937 25577 1.483 941 11933 1.371 947 140159 1.729 953 26203 1.483 967 170921 1.753 971 27827 1.488 977 12157 1.366 983 252727 1.805 991 2011 1.103 997 37813 1.527 The question was not about the third sentence (which is obvious), but about the last one. Sorry for my bad English.
2005-07-26, 12:49   #20
AntonVrba

Jun 2005

2×72 Posts

Quote:
 Originally Posted by alpertron The question was not about the third sentence (which is obvious), but about the last one. Sorry for my bad English.

Quote:
 Originally Posted by alpertron Since p# + Q does not have any divisor less than or equal to p, the probability that p# + Q is prime should be much greater than T + Q when T is a random number near p#. The same argument can be said for p# - Q..
Yes, I believe that centred around p# a region of prime rich clusters and the large gap conjectured by Cramer will not be found here.

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