2023-01-10, 18:11 | #1 |
Feb 2004
France
2×7×67 Posts |
DiGraph under x^2-2 modulo a Wagstaff number
I have written a C program which computes the cycles of the DiGraph under x^2-2 modulo the Wagstaff numbers with q prime from 7 to 31 (Only W29 is not prime).
The results are: Code:
q: 7 ---------------------------- length number 1 2 | 2 -> ... 3 1 | 8 -> ... 5 1 | 4 -> ... 6 1 | 23 -> ... q: 11 ---------------------------- length number 1 2 | 2 -> ... 3 1 | 211 -> ... 5 4 | 223 -> ... 9 9 | 47 -> ... 10 15 | 14 -> ... q: 13 ---------------------------- length number 1 2 | 2 -> ... 2 1 | 755 -> ... 3 1 | 1758 -> ... 4 1 | 1074 -> ... 6 6 | 18 -> ... 11 31 | 4 -> ... 12 53 | 3 -> ... q: 17 ---------------------------- length number 1 2 | 2 -> ... 2 1 | 4906 -> ... 4 2 | 21843 -> ... 5 3 | 37607 -> ... 8 20 | 127 -> ... 15 363 | 527 -> ... 16 672 | 3 -> ... q: 19 ---------------------------- length number 1 2 | 2 -> ... 3 2 | 138706 -> ... 6 4 | 4861 -> ... 9 37 | 171 -> ... 17 1285 | 47 -> ... 18 2407 | 23 -> ... q: 23 ---------------------------- length number 1 2 | 2 -> ... 7 9 | 2529946 -> ... 11 124 | 1292767 -> ... 21 16641 | 47 -> ... 22 31713 | 14 -> ... q: 29 ---------------------------- length number 1 4 | 2 -> ... 2 6 | 24924406 -> ... 3 2 | 63039968 -> ... 4 4 | 139525776 -> ... 6 1 | 76119878 -> ... 12 9 | 56973630 -> ... 14 22 | 89478483 -> ... 28 24 | 154455222 -> ... 42 1 | 290653 -> ... 84 2 | 70628741 -> ... 363 18 | 12228457 -> ... 726 9 | 105518052 -> ... 1452 9 | 20984038 -> ... 5082 9 | 69754008 -> ... 10164 198 | 47 -> ... 21665 2 | 5299202 -> ... 43330 8 | 91807 -> ... 86660 23 | 1022 -> ... 129990 18 | 254 -> ... 259980 46 | 23 -> ... q: 31 ---------------------------- length number 1 2 | 2 -> ... 3 1 | 288941458 -> ... 5 6 | 32768 -> ... 6 1 | 79007128 -> ... 10 48 | 12425 -> ... 15 1454 | 141681407 -> ... 29 3085465 | 47 -> ... 30 5964488 | 23 -> ... When Wq is prime, the length always divides either q-1 or q-2. Where "X -> ..." is an example of such a cycle, giving the first element of the cycle. It appears that the number of cycles for length equal to q-2 and q-1 is given by the OEIS list A165921 : Code:
n a(n) ... 5 1 6 1 ... 9 9 10 15 11 31 12 53 ... 15 363 16 672 17 1285 18 2407 ... 21 16641 22 31713 ... 29 3085465 30 5964488 Code:
-1 mod 4 : q=7, 11, 19, 23, 31 3 -> 7 -> 47 -- q-2 -> 47 +1 mod 4 : q=13, 17, 29 3 -- q-1 -> 3 -1 mod 6 : q=11, 17, 23, 29 4 -> 14 -- q-1 -> 14 5 -> 23 -> 527 -- q-2 -> 527 +1 mod 6 : q=13, 19, 31 4 -- q-2 -> 4 5 -> 23 -- q-1 -> 23 More interesting are the three following universal seeds (in addition to 3/2 (or 1/4) used by Gerbicz): Code:
S0=1154: 6 -> 34 -> 1154 -- q-3 -> -34 -> 1154 Code:
S0=23/8 mod Wq S0=Mersenne(q-3)=2^(q-3)-1 Code:
S0=5^2+1/5^2 (Using S0=k^2+1/k^2 fits with several papers I've read.) I've checked it up to q=3539. Moreover, we have: Code:
for n=1 ... q-1 : 5^(2^n) + 1/5^(2^n) = S(n-1) modulo Wq About q=29, please note that there are 24 cycles of length 28, and none of length 27. None of these cycles of length 28 run into the known universal seeds 1154, 3/2, 23/8, M(q-3), or 5^2+1/5^2 mod Wq. Here is some code for checking validity of the seeds: Code:
Cycle of length q-2: forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(1154,w); s=s0;for(i=1,q-2,s=Mod(s^2-2,w));if(s==s0,print(q))) Cycles of length q-1: forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(23/8,w); s=s0;for(i=1,q-1,s=Mod(s^2-2,w));if(s==s0,print(q))) forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(5^2+1/5^2,w); s=s0;for(i=1,q-1,s=Mod(s^2-2,w));if(s==s0,print(q))) forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(2^(q-3)-1,w); s=s0;for(i=1,q-1,s=Mod(s^2-2,w));if(s==s0,print(q))) forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(3/2,w); s=s0;for(i=1,q-1,s=Mod(s^2-2,w));if(s==s0,print(q))) - When a Wagstaff number is a prime, its DiGraph shows perfect symetries and properties. - When a Wagstaff number is not a prime, its DiGraph shows some mess. Thus, how can we use such information for building the "sufficiency" part of the proof of our conjectures ? How can we prove that, if Wq is not a prime, then the property (use one of the cycles starting with a universal seed) does not stand ? Which theory/tools can we use ? Last fiddled with by T.Rex on 2023-01-10 at 18:17 |
2023-01-10, 21:12 | #2 |
Feb 2004
France
2×7×67 Posts |
I forgot to talk about the universal seed for q-1 cycle found by kijinSeija: W(q-2).
However, as paulunderwood has shown, it's the same as 1/4 mod Wq. And (3/2)^2-2 mod Wq = 1/4 Mod Wq. Note that kijinSeija also found that q^2 is a universal seed for Wagstaff under x2. Code:
forprime(q=5,3539,w=(2^q+1)/3;s0=Mod(q^2,w);s=s0;for(i=1,q-1,s=Mod(s^2,w));if(s==s0,print(q))) |
2023-01-16, 09:40 | #3 |
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
13757_{8} Posts |
How do the cycle lengths behave for other composite exponents?
I wonder whether the proportion of very short cycle lengths could be different for composite exponents compared with prime exponents. If so, it may be possible to screen candidates based on checking a few random seeds for length. To check this, it shouldn't be necessary to calculate all the seeds for larger exponents. A random 1M would be enough(and make comparison easier without calculating %s) Would be happy to fiddle with your c code if you make it available. Is there any logic behind the missing cycle lengths? Looking at it, there seem to be patterns, but they often don't quite hold. Maybe this would be clearer for larger exponents? |
2023-02-22, 17:14 | #4 |
Feb 2004
France
2×7×67 Posts |
Another possible way to find Wagstaff ( PRPs, using a Cycle of length of the DiGraph under .
Let: . We have: . It seems that: is (probably) prime iff . Which is equivallent to say: is (probably) prime iff . The following PARI/gp code checks the above test for all q primes from 11 to 127031. If the test is sucessful and q belongs to the list of known primes p such that is prime, then: "q +" is printed. If the test is sucessful and is NOT prime, then "q --" is printed. Code:
L = [5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031] f(S0)={forprime(q=11,127031, W=(2^q+1)/3; S=S0; found=0; for(i=1,q-2, S=Mod(S^2-2,W)); if(lift(S)==S0, for(j=1,length(L), if(q==L[j],found=1;break)); if(found==1,print(q," +"),print(q," --"))))} f(1154) 13 + 17 + 19 + 23 + 31 + 43 + 61 + .... |
2023-02-28, 11:14 | #5 |
Feb 2004
France
3AA_{16} Posts |
A French mathematician and I are working on this subject.
He has built a formula for computing the length of the cycles. He will very probably publish his work in the next future. Here are the results I've computed for p prime < 200. Wagstaff primes: Code:
p= 13: [1, 2, 3, 4, 6, 11, 12] p= 17: [1, 2, 4, 5, 8, 15, 16] p= 19: [1, 3, 6, 9, 17, 18] p= 23: [1, 7, 11, 21, 22] p= 31: [1, 3, 5, 6, 10, 15, 29, 30] p= 43: [1, 3, 6, 7, 14, 21, 41, 42] p= 61: [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 59, 60] p= 79: [1, 3, 6, 7, 11, 13, 26, 39, 77, 78] p=101: [1, 2, 5, 9, 10, 11, 20, 25, 33, 50, 99, 100] p=127: [1, 3, 5, 6, 7, 9, 14, 18, 21, 25, 42, 63, 125, 126] p=167: [1, 5, 11, 15, 33, 55, 83, 165, 166] p=191: [1, 5, 7, 9, 10, 19, 21, 27, 38, 63, 95, 189, 190] p=199: [1, 3, 6, 9, 11, 18, 22, 33, 66, 99, 197, 198] Code:
p= 29: [1, 2, 3, 4, 6, 12, 14, 28, 42, 84, 363, 726, 1452, 5082, 10164, 21665, 43330, 86660, 129990, 259980] ... n= 20 p= 37: [1, 3, 7, 18, 21, 36, 126, 252, 8221, 24663, 57547, 58065, 147978, 172641, 295956, 348390, 696780] ... n= 17 p= 41: [1, 3, 6, 10, 11, 20, 30, 33, 60, 66, 110, 190, 570, 4180, 12540, 42047, 83102, 126141, 166204, 249306, 252282, 415510, 420470, 498612, 831020, 840940, 925034, 1246530, 2493060, 2522820, ... n= 34 p= 47: [1, 2, 3, 6, 23, 35, 39, 46, 69, 70, 78, 83, 92, 105, 138, 210, 498, 805, 897, 1365, 1443, 1794, 1909, 2545, 2730, 2905, 3220, 3818, 5772, 6474, ... n= 85 p= 53: [1, 5, 10, 20, 26, 52, 130, 260, 33627438, 67254876, 168137190, 336274380, 637722971873, 1275445943746, 3188614859365, 6377229718730] ... n= 16 p= 59: [1, 2, 3, 4, 6, 8, 12, 18, 24, 29, 36, 39, 58, 72, 78, 87, 116, 156, 174, 232, 234, 312, 348, 468, 522, 696, 936, 1044, 1131, 2088, ... n= 128 p= 67: [1, 2, 3, 5, 6, 10, 15, 30, 33, 66, 130, 165, 330, 390, 2559, 4195, 4290, 5118, 8390, 10236, 12585, 12795, 16780, 25170, 25590, 28149, 35229, 50340, 51180, 56298, ... n= 79 p= 71: [1, 2, 3, 6, 35, 70, 105, 140, 210, 1839, 3678, 8754, 17508, 64365, 128730, 257460, 306390, 612780, 2371671, 4743342, 5366202, 7051205, 10732404, 14102410, 21153615, 28204820, 42307230, 76763854, 83008485, 166016970, ... n= 51 p= 73: [1, 9, 18, 135, 270, 438, 1314, 1547, 13923, 19710, 27846, 208845, 417690, 677586, 2032758, 30491370, 51060630, 102121260, 153181890, 306363780, 919091340, 3727425990, 7454851980, 22364555940, 67093667820, 157981589220, 473944767660, 1421834302980, 11532656013060, 34597968039180, ... n= 34 p= 83: [1, 2, 3, 6, 12, 24, 41, 48, 82, 123, 164, 221, 246, 442, 492, 663, 984, 1326, 1968, 2652, 5304, 9061, 9711, 10608, 13973, 18122, 19422, 20061, 27183, 27946, ... n= 216 p= 89: [1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 30, 33, 44, 60, 66, 88, 98, 120, 132, 179, 189, 196, 264, 294, 330, 358, 378, 392, 537, 588, ... n= 522 p= 97: [1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 40, 48, 60, 72, 77, 90, 120, 144, 154, 180, 231, 240, 308, 360, 462, 616, 720, ... n= 131 p=103: [1, 2, 5, 6, 20, 35, 51, 60, 70, 102, 140, 204, 210, 255, 420, 622, 682, 1020, 1244, 1364, 1785, 3570, 7140, 21770, 23870, 31722, 34782, 43540, 47740, 63444, ... n= 135 p=107: [1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 53, 66, 88, 106, 132, 159, 212, 264, 318, 424, 583, 636, 1103, 1119, 1272, 1749, 2206, 2238, ... n= 138 p=109: [1, 2, 3, 4, 6, 8, 12, 14, 18, 24, 26, 28, 36, 42, 48, 52, 72, 78, 84, 104, 126, 144, 156, 168, 182, 234, 252, 312, 336, 364, ... n= 312 p=113: [1, 2, 3, 4, 5, 6, 9, 10, 12, 14, 15, 18, 20, 28, 30, 36, 39, 42, 45, 60, 70, 78, 84, 90, 117, 126, 140, 156, 158, 180, ... n= 978 p=131: [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 37, 42, 60, 65, 74, 84, 111, 130, 148, 195, 202, 222, 251, 260, 370, ... n= 438 p=137: [1, 2, 4, 5, 8, 9, 10, 18, 20, 30, 34, 35, 36, 40, 45, 60, 68, 70, 72, 90, 120, 136, 140, 170, 180, 210, 221, 280, 306, 340, ... n= 1142 p=139: [1, 2, 3, 4, 5, 6, 7, 12, 15, 20, 21, 26, 28, 30, 35, 42, 48, 52, 60, 69, 70, 78, 84, 96, 105, 138, 140, 210, 240, 276, ... n= 775 p=149: [1, 3, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 21, 24, 29, 30, 35, 36, 40, 42, 45, 48, 51, 60, 63, 70, 74, 79, 80, 83, 84, 87, 90, 99, 102, 105, 120, 126, 145, 148, 153, 166 ... n= 2444 p=151: [1, 2, 5, 9, 10, 11, 15, 18, 22, 23, 30, 45, 46, 47, 55, 65, 90, 94, 110, 115, 130, 165, 195, 198, 207, 230, 235, 253, 330, 345, ... n= 439 p=157: [1, 2, 3, 4, 6, 11, 12, 14, 22, 26, 28, 30, 33, 42, 44, 52, 60, 66, 78, 83, 84, 88, 104, 105, 132, 142, 154, 156, 166, 182, ... n= 3220 p=163: [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 23, 24, 30, 40, 46, 60, 69, 81, 92, 120, 138, 143, 162, 184, 209, 230, 276, 286, 324, ... n= 1575 p=173: [1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15, 20, 22, 28, 30, 33, 42, 44, 55, 60, 66, 70, 84, 86, 110, 132, 140, 154, 165, 172, , 210, 220, 233, ... n= 4312 p=179: [1, 2, 4, 9, 14, 18, 22, 28, 30, 36, 44, 60, 89, 90, 126, 154, 178, 179, 180, 210, 252, 308, 323, 330, 356, 386, 420, 630, 660, 772, ... n= 378 p=181: [1, 2, 3, 6, 15, 30, 90, 180, 4861, 9722, 11543, 23086, 34629, 69258, 72915, 145830, 173145, 206493, 346290, 412986, 437490, 874980, 1032465, 1038870, 2064930, 6194790, 12389580, 13883549, 23834330, 27767098, ... n= 331 p=193: [1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 48, 55, 60, 65, 66, 96, 110, 130, 132, 165, 195, 220, 240, 260, 273, ... n= 5354 p=197: [1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 20, 21, 24, 28, 29, 30, 35, 36, 45, 60, 63, 72, 83, 84, 87, 90, 98, 105, 116, 120, 140, 145, 166, 168, 180, 196, 210, 249, ... n= 6565 Last fiddled with by T.Rex on 2023-02-28 at 11:18 |
2023-03-02, 10:01 | #6 |
Feb 2004
France
2·7·67 Posts |
Here are additionnal results for Wp non-primes.
It is noticeable that p-2 appears as a length of cycles for p=331 (but no p-1 cycle). So, the conclusion seems to be: p-1 appears often but not always, p-2 appears rarely, both p-1 and p-2 often do not appear. thus: no clear rule for non-primes Wp ! Code:
p=211: [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 23, 25, 28, 30, 33, 35, 36, 42, 44, 45, 46, 50, 55, 60, 63, 66, 69, 70, 75, 77, 84, 90, 92, 99, 100, 105, 110, 115, 126, 132, 138, 140, 142, 150, 154, 161, 165, 175, 180, 186, 190, 198, 210, 220, 225, 230, 231, 252, ...] p=223: [1, 2, 24, 29, 37, 48, 58, 74, 98, 146, 148, 196, 888, 1073, 1752, 1776, 2146, ...] p=227: [1, 2, 4, 38, 65, 76, 113, 130, 226, 260, 299, 452, 491, 598, ...] p=229: [1, 5, 10, 18, 36, 38, 180, 190, 342, 380, 684, 3420, 20474, ...] p=233: [1, 2, 4, 5, 6, 10, 12, 18, 20, 29, 30, 36, 58, 60, 90, 116, 145, 174, 180, 290, 348, 371, 522, 580, ...] p=239: [1, 2, 4, 11, 22, 34, 36, 44, 65, 68, 69, 72, 119, 130, 138, 174, 238, 260, 348, 374, 476, 612, ...] p=241: [1, 2, 3, 4, 6, 11, 12, 22, 24, 33, 44, 50, 66, 100, 132, 150, 264, 300, 419, 545, 550, ...] p=251: [1, 2, 3, 5, 6, 9, 10, 11, 15, 22, 23, 25, 30, 32, 36, 45, 46, 50, 55, 64, 75, 96, 99, 100, 110, 115, 160, 180, 192, 198, 207, 225, 230, 253, 275, 300, 320, 352, ...] p=257: [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 50, 60, 80, 100, 120, 150, 174, 200, 240, 300, 348, 398, 400, 473, ...] p=263: [1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 21, 24, 25, 28, 30, 39, 42, 50, 56, 60, 75, 78, 84, 100, 120, 131, 150, 156, 168, 175, 195, 200, 210, 262, 273, 300, 312, 350, 390, ...] p=271: [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 18, 20, 22, 28, 30, 33, 36, 42, 44, 45, 50, 55, 60, 66, 70, 84, 90, 100, 110, 126, 132, 135, 140, 150, 165, 180, 191, 198, 210, 220, 252, 270, 300, 308, 330, 350, 382, ...] p=277: [1, 3, 14, 33, 42, 46, 48, 65, 84, 92, 96, 138, 195, 276, 322, 336, 390, 528, 644, ...] p=281: [1, 2, 4, 23, 26, 35, 46, 70, 92, 140, 598, 805, 834, 910, 1610, ...] p=283: [1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 18, 20, 24, 28, 30, 36, 40, 42, 47, 48, 56, 60, 69, 70, 72, 84, 90, 94, 96, 120, 138, 140, 141, 144, 168, 180, 188, 210, 235, 240, 276, 280, 282, 288, 336, 345, 360, 376, ...] p=307: [1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 48, 51, 56, 60, 69, 70, 84, 96, 102, 105, 120, 138, 140, 168, 204, 210, 240, 250, 255, 266, 276, 280, 336, 345, 357, 408, 420, ...] p=311: [1, 2, 3, 7, 9, 12, 18, 28, 35, 36, 42, 51, 63, 70, 84, 102, 105, 126, 140, 155, 204, 210, 252, 310, 315, 357, 420, 465, 630, ...] p=317: [1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15, 18, 20, 21, 22, 28, 30, 33, 36, 42, 44, 55, 60, 66, 69, 70, 84, 90, 105, 110, 126, 132, 138, 140, 154, 158, 165, 180, 198, 210, 220, 251, 252, 276, 308, 316, 330, 345, 396, 414, 420, ...] p=331: [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 48, 51, 60, 72, 90, 102, 120, 130, 144, 153, 180, 204, 240, 255, 260, 306, 329, 360, 390, 408, 441, 510, ...] |
2023-03-02, 12:19 | #7 |
Feb 2004
France
2·7·67 Posts |
Here is the probably last result for prime:
Code:
p=347: [1, 5, 15, 23, 69, 115, 173, 345, 346] |
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