mersenneforum.org Happy New Year & There are just not enough Numerological Threads
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 2021-01-01, 22:31 #1 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7·283 Posts Happy New Year & There are just not enough Numerological Threads Code:  \\\\DSN-320-A-Pari-GP Code By Rashid Naimi 1/1/2121 >>>>>>>>>>>>>>>Error caught by Krisel. Now I have to reweight all the post dated checks for the year primeCounter=0 compositeCounter=0 LowP=101 highP=-1 for(n=1,19^19,{\\\\\\\\\\\\\\\\\\\\\\\\\\ n1=2*n; n2=(2+n*2)*n/2; addend0=6; m0=3; addend1=(n2+1-m0)*2+addend0; addend2=addend1+1; p=addend1*2+1; lowerFactor1=(p-1)/2; upperFactor1 = lowerFactor1+1; mLower1 = Mod(lowerFactor1,p); mLower1 = mLower1 *lowerFactor1; mUpper1 = Mod(upperFactor1,p); mUpper1 = mUpper1*upperFactor1; lowerFactor1=lowerFactor1+1; upperFactor1 = upperFactor1-1; primeFlag=1; theDepth=1+2^14; \\\\\\ the larger this value, the less composites will pass the test. if(theDepth>(p-1)/2,theDepth=(p-1)/2); for(j=1,theDepth, lowerFactor1=lowerFactor1-1; upperFactor1 = upperFactor1+1; \\mLower1 = mLower1/lowerFactor1; iferr(mLower1 = mLower1/lowerFactor1, E, mLower1 =0;primeFlag=0;); \\mUpper1 = mUpper1/upperFactor1; iferr(mUpper1 = mUpper1/upperFactor1, E, mUpper1 =0;primeFlag=0;); ); if(!primeFlag, next(1); , print("\nn = ",n); print("n1 = ",n1); print("n2 = ",n2); print("p = ",p," >> ", ispseudoprime(p)); print("**** ",LowP,"% < ",prcntg,"% < ",highP,"% Prime!"); ); if(!ispseudoprime(p), if(primeFlag, \\next(19); \\Uncomment this line to abort on 1st test failiure ); ); if(ispseudoprime(p),primeCounter=primeCounter+1,if(primeFlag,compositeCounter=compositeCounter+1)); prcntg =round(primeCounter/(primeCounter+compositeCounter)*100); if(prcntg < LowP,LowP=prcntg); if(prcntg > highP,highP=prcntg); }) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\^^^^^^^^^^^^^^^^^^ I will let you figure out why it does what it does. I am only off by 100 years. ETA: Hint #1: It is based on Wilson's Theorem: https://en.wikipedia.org/wiki/Wilson%27s_theorem Last fiddled with by a1call on 2021-01-01 at 23:23
 2021-01-02, 00:48 #2 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7×283 Posts Hint #2: For all positive integers p where valuation(p-1,2)==1 , Mod( ((p-1)/2)! , p) == 1 || Mod( ((p-1)/2)! , p-1) if and only if p is prime Hint #3: For all primes p where valuation(p-1,2)>1 , there exist coprime integers a and b such that a+b=p && p | ab-1 (Not all but some composites satisfy this condition as well such as 25 where a=7 and b= 18) ETA: being off by 100 years isn't that bad really, considering that numbers are unlimited and I could potentially be off by an indefinitely larger value. ETA II Hint #4: For all primes p where valuation(p-1,2)>1 , There are coprime integers a and b such that Mod(( (p-1)/2)!, p) == a and Mod(( (p-2)! / (p-1)/2)!, p) == b and a+b=p and 1
 2021-01-02, 01:56 #3 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7·283 Posts Re: Hint #4 if m=Mod(a,p) Unless p is prime, iterations of m=m/c where c iterates from (p-1)/2 to 2 or vise versa will definitely fail to evaluate at some point which is what the code checks. Thank you for your time. Last fiddled with by a1call on 2021-01-02 at 02:05
 2021-01-02, 05:16 #4 ONeil   Dec 2017 24·3·5 Posts Hey at1call will the above code predict if a large number is prime or composite effectively and fast?
 2021-01-02, 05:29 #5 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7×283 Posts Nah, unfortunately you will have to set theDepth parameter very high for large numbers which will make the process unusable for larger numbers. That is unless someone can figure out the mechanics of when m/c fails to evaluate and predict when it will happen. I gave up trying to do that. Last fiddled with by a1call on 2021-01-02 at 05:30
 2021-01-05, 06:41 #6 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7×283 Posts So it turns out it's just trial factoring. Mod(a,p)/k will fail to evaluate if k has a prime cofactor with p.
 2021-01-12, 05:25 #7 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7BD16 Posts Some more insights from the future (just kidding). So it turns out that the number of (a, b) pairs for each n is a function of number of prime factors of n. If n is prime or otherwise a power of a prime there is only one pair of positive integers (a, b) satisfying the condition: a+b=n and n | ab-1 Where valuation(n-1,2) > 1 --------------------- If n has 2 prime factors there will be 2 pairs of (a, b)'s if n has 3 prime factors there will be 4. pairs of (a, b)'s .... Furthermore the differences of multipair (a, b)'s from different pairs will have a common factor with n. so for n =65 (a, b)1 = ( 8, 57) (a, b)2 = (18, 47) and gcd(18-8,65) == gcd(47-57,65) = 5 gcd(57-18,65) == gcd(47-8,65) = 13 where 65 = 5*13 Code: \\DSZ-100-A by Rashid Naimi 1/12/2221 forstep(n=3,19^3,2,{ for(a=2,(n-1)/2, b=n-a; m=lift(Mod(a*b,n)); if(m==1, print("\n",n," >> ",factor(n)," >> ",isprime(n)); print(a,", ",b); ); ); }) Output: Code:  5 >> Mat([5, 1]) >> 1 2, 3 13 >> Mat([13, 1]) >> 1 5, 8 17 >> Mat([17, 1]) >> 1 4, 13 25 >> Mat([5, 2]) >> 0 7, 18 29 >> Mat([29, 1]) >> 1 12, 17 37 >> Mat([37, 1]) >> 1 6, 31 41 >> Mat([41, 1]) >> 1 9, 32 53 >> Mat([53, 1]) >> 1 23, 30 61 >> Mat([61, 1]) >> 1 11, 50 65 >> [5, 1; 13, 1] >> 0 8, 57 65 >> [5, 1; 13, 1] >> 0 18, 47 73 >> Mat([73, 1]) >> 1 27, 46 85 >> [5, 1; 17, 1] >> 0 13, 72 85 >> [5, 1; 17, 1] >> 0 38, 47 89 >> Mat([89, 1]) >> 1 34, 55 97 >> Mat([97, 1]) >> 1 22, 75 101 >> Mat([101, 1]) >> 1 10, 91 109 >> Mat([109, 1]) >> 1 33, 76 113 >> Mat([113, 1]) >> 1 15, 98 125 >> Mat([5, 3]) >> 0 57, 68 137 >> Mat([137, 1]) >> 1 37, 100 145 >> [5, 1; 29, 1] >> 0 12, 133 145 >> [5, 1; 29, 1] >> 0 17, 128 149 >> Mat([149, 1]) >> 1 44, 105 157 >> Mat([157, 1]) >> 1 28, 129 169 >> Mat([13, 2]) >> 0 70, 99 173 >> Mat([173, 1]) >> 1 80, 93 181 >> Mat([181, 1]) >> 1 19, 162 185 >> [5, 1; 37, 1] >> 0 43, 142 185 >> [5, 1; 37, 1] >> 0 68, 117 193 >> Mat([193, 1]) >> 1 81, 112 197 >> Mat([197, 1]) >> 1 14, 183 205 >> [5, 1; 41, 1] >> 0 32, 173 205 >> [5, 1; 41, 1] >> 0 73, 132 221 >> [13, 1; 17, 1] >> 0 21, 200 221 >> [13, 1; 17, 1] >> 0 47, 174 229 >> Mat([229, 1]) >> 1 107, 122 233 >> Mat([233, 1]) >> 1 89, 144 241 >> Mat([241, 1]) >> 1 64, 177 257 >> Mat([257, 1]) >> 1 16, 241 265 >> [5, 1; 53, 1] >> 0 23, 242 265 >> [5, 1; 53, 1] >> 0 83, 182 269 >> Mat([269, 1]) >> 1 82, 187 277 >> Mat([277, 1]) >> 1 60, 217 281 >> Mat([281, 1]) >> 1 53, 228 289 >> Mat([17, 2]) >> 0 38, 251 293 >> Mat([293, 1]) >> 1 138, 155 305 >> [5, 1; 61, 1] >> 0 72, 233 305 >> [5, 1; 61, 1] >> 0 133, 172 313 >> Mat([313, 1]) >> 1 25, 288 317 >> Mat([317, 1]) >> 1 114, 203 325 >> [5, 2; 13, 1] >> 0 18, 307 325 >> [5, 2; 13, 1] >> 0 57, 268 337 >> Mat([337, 1]) >> 1 148, 189 349 >> Mat([349, 1]) >> 1 136, 213 353 >> Mat([353, 1]) >> 1 42, 311 365 >> [5, 1; 73, 1] >> 0 27, 338 365 >> [5, 1; 73, 1] >> 0 173, 192 373 >> Mat([373, 1]) >> 1 104, 269 377 >> [13, 1; 29, 1] >> 0 70, 307 377 >> [13, 1; 29, 1] >> 0 99, 278 389 >> Mat([389, 1]) >> 1 115, 274 397 >> Mat([397, 1]) >> 1 63, 334 401 >> Mat([401, 1]) >> 1 20, 381 409 >> Mat([409, 1]) >> 1 143, 266 421 >> Mat([421, 1]) >> 1 29, 392 425 >> [5, 2; 17, 1] >> 0 132, 293 425 >> [5, 2; 17, 1] >> 0 157, 268 433 >> Mat([433, 1]) >> 1 179, 254 445 >> [5, 1; 89, 1] >> 0 123, 322 445 >> [5, 1; 89, 1] >> 0 212, 233 449 >> Mat([449, 1]) >> 1 67, 382 457 >> Mat([457, 1]) >> 1 109, 348 461 >> Mat([461, 1]) >> 1 48, 413 481 >> [13, 1; 37, 1] >> 0 31, 450 481 >> [13, 1; 37, 1] >> 0 216, 265 485 >> [5, 1; 97, 1] >> 0 22, 463 485 >> [5, 1; 97, 1] >> 0 172, 313 493 >> [17, 1; 29, 1] >> 0 157, 336 493 >> [17, 1; 29, 1] >> 0 191, 302 505 >> [5, 1; 101, 1] >> 0 192, 313 505 >> [5, 1; 101, 1] >> 0 212, 293 509 >> Mat([509, 1]) >> 1 208, 301 521 >> Mat([521, 1]) >> 1 235, 286 533 >> [13, 1; 41, 1] >> 0 73, 460 533 >> [13, 1; 41, 1] >> 0 255, 278 541 >> Mat([541, 1]) >> 1 52, 489 545 >> [5, 1; 109, 1] >> 0 33, 512 545 >> [5, 1; 109, 1] >> 0 142, 403 557 >> Mat([557, 1]) >> 1 118, 439 565 >> [5, 1; 113, 1] >> 0 98, 467 565 >> [5, 1; 113, 1] >> 0 128, 437 569 >> Mat([569, 1]) >> 1 86, 483 577 >> Mat([577, 1]) >> 1 24, 553 593 >> Mat([593, 1]) >> 1 77, 516 601 >> Mat([601, 1]) >> 1 125, 476 613 >> Mat([613, 1]) >> 1 35, 578 617 >> Mat([617, 1]) >> 1 194, 423 625 >> Mat([5, 4]) >> 0 182, 443 629 >> [17, 1; 37, 1] >> 0 191, 438 629 >> [17, 1; 37, 1] >> 0 302, 327 641 >> Mat([641, 1]) >> 1 154, 487 653 >> Mat([653, 1]) >> 1 149, 504 661 >> Mat([661, 1]) >> 1 106, 555 673 >> Mat([673, 1]) >> 1 58, 615 677 >> Mat([677, 1]) >> 1 26, 651 685 >> [5, 1; 137, 1] >> 0 37, 648 685 >> [5, 1; 137, 1] >> 0 237, 448 689 >> [13, 1; 53, 1] >> 0 83, 606 689 >> [13, 1; 53, 1] >> 0 242, 447 697 >> [17, 1; 41, 1] >> 0 132, 565 697 >> [17, 1; 41, 1] >> 0 319, 378 701 >> Mat([701, 1]) >> 1 135, 566 709 >> Mat([709, 1]) >> 1 96, 613 725 >> [5, 2; 29, 1] >> 0 157, 568 725 >> [5, 2; 29, 1] >> 0 307, 418 733 >> Mat([733, 1]) >> 1 353, 380 745 >> [5, 1; 149, 1] >> 0 193, 552 745 >> [5, 1; 149, 1] >> 0 342, 403 757 >> Mat([757, 1]) >> 1 87, 670 761 >> Mat([761, 1]) >> 1 39, 722 769 >> Mat([769, 1]) >> 1 62, 707 773 >> Mat([773, 1]) >> 1 317, 456 785 >> [5, 1; 157, 1] >> 0 28, 757 785 >> [5, 1; 157, 1] >> 0 342, 443 793 >> [13, 1; 61, 1] >> 0 255, 538 793 >> [13, 1; 61, 1] >> 0 294, 499 797 >> Mat([797, 1]) >> 1 215, 582 809 >> Mat([809, 1]) >> 1 318, 491 821 >> Mat([821, 1]) >> 1 295, 526 829 >> Mat([829, 1]) >> 1 246, 583 841 >> Mat([29, 2]) >> 0 41, 800 845 >> [5, 1; 13, 2] >> 0 268, 577 845 >> [5, 1; 13, 2] >> 0 408, 437 853 >> Mat([853, 1]) >> 1 333, 520 857 >> Mat([857, 1]) >> 1 207, 650 865 >> [5, 1; 173, 1] >> 0 93, 772 865 >> [5, 1; 173, 1] >> 0 253, 612 877 >> Mat([877, 1]) >> 1 151, 726 881 >> Mat([881, 1]) >> 1 387, 494 901 >> [17, 1; 53, 1] >> 0 30, 871 901 >> [17, 1; 53, 1] >> 0 242, 659 905 >> [5, 1; 181, 1] >> 0 162, 743 905 >> [5, 1; 181, 1] >> 0 343, 562 925 >> [5, 2; 37, 1] >> 0 43, 882 925 >> [5, 2; 37, 1] >> 0 68, 857 929 >> Mat([929, 1]) >> 1 324, 605 937 >> Mat([937, 1]) >> 1 196, 741 941 >> Mat([941, 1]) >> 1 97, 844 949 >> [13, 1; 73, 1] >> 0 265, 684 949 >> [13, 1; 73, 1] >> 0 411, 538 953 >> Mat([953, 1]) >> 1 442, 511 965 >> [5, 1; 193, 1] >> 0 112, 853 965 >> [5, 1; 193, 1] >> 0 467, 498 977 >> Mat([977, 1]) >> 1 252, 725 985 >> [5, 1; 197, 1] >> 0 183, 802 985 >> [5, 1; 197, 1] >> 0 408, 577 997 >> Mat([997, 1]) >> 1 161, 836 1009 >> Mat([1009, 1]) >> 1 469, 540 1013 >> Mat([1013, 1]) >> 1 45, 968 1021 >> Mat([1021, 1]) >> 1 374, 647 1025 >> [5, 2; 41, 1] >> 0 32, 993 1025 >> [5, 2; 41, 1] >> 0 132, 893 1033 >> Mat([1033, 1]) >> 1 355, 678 1037 >> [17, 1; 61, 1] >> 0 72, 965 1037 >> [17, 1; 61, 1] >> 0 438, 599 1049 >> Mat([1049, 1]) >> 1 426, 623 1061 >> Mat([1061, 1]) >> 1 103, 958 1069 >> Mat([1069, 1]) >> 1 249, 820 1073 >> [29, 1; 37, 1] >> 0 191, 882 1073 >> [29, 1; 37, 1] >> 0 302, 771 1093 >> Mat([1093, 1]) >> 1 530, 563 1097 >> Mat([1097, 1]) >> 1 341, 756 1105 >> [5, 1; 13, 1; 17, 1] >> 0 47, 1058 1105 >> [5, 1; 13, 1; 17, 1] >> 0 242, 863 1105 >> [5, 1; 13, 1; 17, 1] >> 0 268, 837 1105 >> [5, 1; 13, 1; 17, 1] >> 0 463, 642 1109 >> Mat([1109, 1]) >> 1 354, 755 1117 >> Mat([1117, 1]) >> 1 214, 903 1129 >> Mat([1129, 1]) >> 1 168, 961 1145 >> [5, 1; 229, 1] >> 0 107, 1038 1145 >> [5, 1; 229, 1] >> 0 122, 1023 1153 >> Mat([1153, 1]) >> 1 140, 1013 1157 >> [13, 1; 89, 1] >> 0 34, 1123 1157 >> [13, 1; 89, 1] >> 0 411, 746 1165 >> [5, 1; 233, 1] >> 0 322, 843 1165 >> [5, 1; 233, 1] >> 0 377, 788 1181 >> Mat([1181, 1]) >> 1 243, 938 1189 >> [29, 1; 41, 1] >> 0 278, 911 1189 >> [29, 1; 41, 1] >> 0 360, 829 1193 >> Mat([1193, 1]) >> 1 186, 1007 1201 >> Mat([1201, 1]) >> 1 49, 1152 1205 >> [5, 1; 241, 1] >> 0 177, 1028 1205 >> [5, 1; 241, 1] >> 0 418, 787 1213 >> Mat([1213, 1]) >> 1 495, 718 1217 >> Mat([1217, 1]) >> 1 78, 1139 1229 >> Mat([1229, 1]) >> 1 597, 632 1237 >> Mat([1237, 1]) >> 1 546, 691 1241 >> [17, 1; 73, 1] >> 0 319, 922 1241 >> [17, 1; 73, 1] >> 0 557, 684 1249 >> Mat([1249, 1]) >> 1 585, 664 1261 >> [13, 1; 97, 1] >> 0 216, 1045 1261 >> [13, 1; 97, 1] >> 0 463, 798 1277 >> Mat([1277, 1]) >> 1 113, 1164 1285 >> [5, 1; 257, 1] >> 0 273, 1012 1285 >> [5, 1; 257, 1] >> 0 498, 787 1289 >> Mat([1289, 1]) >> 1 479, 810 1297 >> Mat([1297, 1]) >> 1 36, 1261 1301 >> Mat([1301, 1]) >> 1 51, 1250 1313 >> [13, 1; 101, 1] >> 0 515, 798 1313 >> [13, 1; 101, 1] >> 0 616, 697 1321 >> Mat([1321, 1]) >> 1 257, 1064 1325 >> [5, 2; 53, 1] >> 0 182, 1143 1325 >> [5, 2; 53, 1] >> 0 507, 818 1345 >> [5, 1; 269, 1] >> 0 82, 1263 1345 >> [5, 1; 269, 1] >> 0 187, 1158 1361 >> Mat([1361, 1]) >> 1 614, 747 1369 >> Mat([37, 2]) >> 0 117, 1252 1373 >> Mat([1373, 1]) >> 1 668, 705 1381 >> Mat([1381, 1]) >> 1 366, 1015 1385 >> [5, 1; 277, 1] >> 0 217, 1168 1385 >> [5, 1; 277, 1] >> 0 337, 1048 1405 >> [5, 1; 281, 1] >> 0 53, 1352 1405 >> [5, 1; 281, 1] >> 0 228, 1177 1409 >> Mat([1409, 1]) >> 1 452, 957 1417 >> [13, 1; 109, 1] >> 0 294, 1123 1417 >> [13, 1; 109, 1] >> 0 512, 905 1429 >> Mat([1429, 1]) >> 1 620, 809 1433 >> Mat([1433, 1]) >> 1 542, 891 1445 >> [5, 1; 17, 2] >> 0 38, 1407 1445 >> [5, 1; 17, 2] >> 0 327, 1118 1453 >> Mat([1453, 1]) >> 1 497, 956 1465 >> [5, 1; 293, 1] >> 0 138, 1327 1465 >> [5, 1; 293, 1] >> 0 448, 1017 1469 >> [13, 1; 113, 1] >> 0 437, 1032 1469 >> [13, 1; 113, 1] >> 0 580, 889 1481 >> Mat([1481, 1]) >> 1 465, 1016 1489 >> Mat([1489, 1]) >> 1 225, 1264 1493 >> Mat([1493, 1]) >> 1 432, 1061 1513 >> [17, 1; 89, 1] >> 0 55, 1458 1513 >> [17, 1; 89, 1] >> 0 123, 1390 1517 >> [37, 1; 41, 1] >> 0 401, 1116 1517 >> [37, 1; 41, 1] >> 0 524, 993 1525 >> [5, 2; 61, 1] >> 0 682, 843 1525 >> [5, 2; 61, 1] >> 0 743, 782 1537 >> [29, 1; 53, 1] >> 0 394, 1143 1537 >> [29, 1; 53, 1] >> 0 447, 1090 1549 >> Mat([1549, 1]) >> 1 88, 1461 1553 >> Mat([1553, 1]) >> 1 339, 1214 1565 >> [5, 1; 313, 1] >> 0 288, 1277 1565 >> [5, 1; 313, 1] >> 0 338, 1227 1585 >> [5, 1; 317, 1] >> 0 203, 1382 1585 >> [5, 1; 317, 1] >> 0 748, 837 1597 >> Mat([1597, 1]) >> 1 610, 987 1601 >> Mat([1601, 1]) >> 1 40, 1561 1609 >> Mat([1609, 1]) >> 1 523, 1086 1613 >> Mat([1613, 1]) >> 1 127, 1486 1621 >> Mat([1621, 1]) >> 1 166, 1455 1625 >> [5, 3; 13, 1] >> 0 57, 1568 1625 >> [5, 3; 13, 1] >> 0 307, 1318 1637 >> Mat([1637, 1]) >> 1 316, 1321 1649 >> [17, 1; 97, 1] >> 0 463, 1186 1649 >> [17, 1; 97, 1] >> 0 701, 948 1657 >> Mat([1657, 1]) >> 1 783, 874 1669 >> Mat([1669, 1]) >> 1 220, 1449 1681 >> Mat([41, 2]) >> 0 378, 1303 1685 >> [5, 1; 337, 1] >> 0 148, 1537 1685 >> [5, 1; 337, 1] >> 0 822, 863 1693 >> Mat([1693, 1]) >> 1 92, 1601 1697 >> Mat([1697, 1]) >> 1 414, 1283 1709 >> Mat([1709, 1]) >> 1 390, 1319 1717 >> [17, 1; 101, 1] >> 0 293, 1424 1717 >> [17, 1; 101, 1] >> 0 616, 1101 1721 >> Mat([1721, 1]) >> 1 473, 1248 1733 >> Mat([1733, 1]) >> 1 410, 1323 1741 >> Mat([1741, 1]) >> 1 59, 1682 1745 >> [5, 1; 349, 1] >> 0 213, 1532 1745 >> [5, 1; 349, 1] >> 0 562, 1183 1753 >> Mat([1753, 1]) >> 1 713, 1040 1765 >> [5, 1; 353, 1] >> 0 42, 1723 1765 >> [5, 1; 353, 1] >> 0 748, 1017 1769 >> [29, 1; 61, 1] >> 0 133, 1636 1769 >> [29, 1; 61, 1] >> 0 621, 1148 1777 >> Mat([1777, 1]) >> 1 775, 1002 1781 >> [13, 1; 137, 1] >> 0 174, 1607 1781 >> [13, 1; 137, 1] >> 0 785, 996 1789 >> Mat([1789, 1]) >> 1 724, 1065 1801 >> Mat([1801, 1]) >> 1 824, 977 1825 >> [5, 2; 73, 1] >> 0 557, 1268 1825 >> [5, 2; 73, 1] >> 0 757, 1068 1853 >> [17, 1; 109, 1] >> 0 251, 1602 1853 >> [17, 1; 109, 1] >> 0 905, 948 1861 >> Mat([1861, 1]) >> 1 61, 1800 1865 >> [5, 1; 373, 1] >> 0 477, 1388 1865 >> [5, 1; 373, 1] >> 0 642, 1223 1873 >> Mat([1873, 1]) >> 1 737, 1136 1877 >> Mat([1877, 1]) >> 1 137, 1740 1885 >> [5, 1; 13, 1; 29, 1] >> 0 278, 1607 1885 >> [5, 1; 13, 1; 29, 1] >> 0 307, 1578 1885 >> [5, 1; 13, 1; 29, 1] >> 0 447, 1438 1885 >> [5, 1; 13, 1; 29, 1] >> 0 853, 1032 1889 >> Mat([1889, 1]) >> 1 331, 1558 1901 >> Mat([1901, 1]) >> 1 218, 1683 1913 >> Mat([1913, 1]) >> 1 712, 1201 1921 >> [17, 1; 113, 1] >> 0 98, 1823 1921 >> [17, 1; 113, 1] >> 0 693, 1228 1933 >> Mat([1933, 1]) >> 1 598, 1335 1937 >> [13, 1; 149, 1] >> 0 44, 1893 1937 >> [13, 1; 149, 1] >> 0 850, 1087 1945 >> [5, 1; 389, 1] >> 0 663, 1282 1945 >> [5, 1; 389, 1] >> 0 893, 1052 1949 >> Mat([1949, 1]) >> 1 589, 1360 1961 >> [37, 1; 53, 1] >> 0 401, 1560 1961 >> [37, 1; 53, 1] >> 0 931, 1030 1973 >> Mat([1973, 1]) >> 1 259, 1714 1985 >> [5, 1; 397, 1] >> 0 63, 1922 1985 >> [5, 1; 397, 1] >> 0 857, 1128 1993 >> Mat([1993, 1]) >> 1 834, 1159 1997 >> Mat([1997, 1]) >> 1 412, 1585 2005 >> [5, 1; 401, 1] >> 0 782, 1223 2005 >> [5, 1; 401, 1] >> 0 822, 1183 2017 >> Mat([2017, 1]) >> 1 229, 1788 2029 >> Mat([2029, 1]) >> 1 992, 1037 2041 >> [13, 1; 157, 1] >> 0 499, 1542 2041 >> [13, 1; 157, 1] >> 0 970, 1071 2045 >> [5, 1; 409, 1] >> 0 143, 1902 2045 >> [5, 1; 409, 1] >> 0 552, 1493 2053 >> Mat([2053, 1]) >> 1 244, 1809 2069 >> Mat([2069, 1]) >> 1 164, 1905 2081 >> Mat([2081, 1]) >> 1 102, 1979 2089 >> Mat([2089, 1]) >> 1 789, 1300 2105 >> [5, 1; 421, 1] >> 0 392, 1713 2105 >> [5, 1; 421, 1] >> 0 813, 1292 2113 >> Mat([2113, 1]) >> 1 65, 2048 2117 >> [29, 1; 73, 1] >> 0 46, 2071 2117 >> [29, 1; 73, 1] >> 0 684, 1433 2125 >> [5, 3; 17, 1] >> 0 557, 1568 2125 >> [5, 3; 17, 1] >> 0 693, 1432 2129 >> Mat([2129, 1]) >> 1 372, 1757 2137 >> Mat([2137, 1]) >> 1 296, 1841 2141 >> Mat([2141, 1]) >> 1 419, 1722 2153 >> Mat([2153, 1]) >> 1 232, 1921 2161 >> Mat([2161, 1]) >> 1 147, 2014 2165 >> [5, 1; 433, 1] >> 0 612, 1553 2165 >> [5, 1; 433, 1] >> 0 687, 1478 2173 >> [41, 1; 53, 1] >> 0 401, 1772 2173 >> [41, 1; 53, 1] >> 0 606, 1567 2197 >> Mat([13, 3]) >> 0 239, 1958 2213 >> Mat([2213, 1]) >> 1 1083, 1130 2221 >> Mat([2221, 1]) >> 1 790, 1431 2225 >> [5, 2; 89, 1] >> 0 568, 1657 2225 >> [5, 2; 89, 1] >> 0 657, 1568 2237 >> Mat([2237, 1]) >> 1 1021, 1216 2245 >> [5, 1; 449, 1] >> 0 67, 2178 2245 >> [5, 1; 449, 1] >> 0 382, 1863 2249 >> [13, 1; 173, 1] >> 0 772, 1477 2249 >> [13, 1; 173, 1] >> 0 785, 1464 2257 >> [37, 1; 61, 1] >> 0 438, 1819 2257 >> [37, 1; 61, 1] >> 0 660, 1597 2269 >> Mat([2269, 1]) >> 1 982, 1287 2273 >> Mat([2273, 1]) >> 1 290, 1983 2281 >> Mat([2281, 1]) >> 1 710, 1571 2285 >> [5, 1; 457, 1] >> 0 348, 1937 2285 >> [5, 1; 457, 1] >> 0 1023, 1262 2293 >> Mat([2293, 1]) >> 1 600, 1693 2297 >> Mat([2297, 1]) >> 1 365, 1932 2305 >> [5, 1; 461, 1] >> 0 48, 2257 2305 >> [5, 1; 461, 1] >> 0 413, 1892 2309 >> Mat([2309, 1]) >> 1 688, 1621 2329 >> [17, 1; 137, 1] >> 0 174, 2155 2329 >> [17, 1; 137, 1] >> 0 922, 1407 2333 >> Mat([2333, 1]) >> 1 108, 2225 2341 >> Mat([2341, 1]) >> 1 153, 2188 2353 >> [13, 1; 181, 1] >> 0 200, 2153 2353 >> [13, 1; 181, 1] >> 0 343, 2010 2357 >> Mat([2357, 1]) >> 1 633, 1724 2377 >> Mat([2377, 1]) >> 1 1134, 1243 2381 >> Mat([2381, 1]) >> 1 69, 2312 2389 >> Mat([2389, 1]) >> 1 285, 2104 2393 >> Mat([2393, 1]) >> 1 971, 1422 2405 >> [5, 1; 13, 1; 37, 1] >> 0 512, 1893 2405 >> [5, 1; 13, 1; 37, 1] >> 0 697, 1708 2405 >> [5, 1; 13, 1; 37, 1] >> 0 993, 1412 2405 >> [5, 1; 13, 1; 37, 1] >> 0 1178, 1227 2417 >> Mat([2417, 1]) >> 1 592, 1825 2425 >> [5, 2; 97, 1] >> 0 507, 1918 2425 >> [5, 2; 97, 1] >> 0 657, 1768 2437 >> Mat([2437, 1]) >> 1 398, 2039 2441 >> Mat([2441, 1]) >> 1 672, 1769 2465 >> [5, 1; 17, 1; 29, 1] >> 0 157, 2308 2465 >> [5, 1; 17, 1; 29, 1] >> 0 302, 2163 2465 >> [5, 1; 17, 1; 29, 1] >> 0 1143, 1322 2465 >> [5, 1; 17, 1; 29, 1] >> 0 1177, 1288 2473 >> Mat([2473, 1]) >> 1 567, 1906 2477 >> Mat([2477, 1]) >> 1 915, 1562 2501 >> [41, 1; 61, 1] >> 0 50, 2451 2501 >> [41, 1; 61, 1] >> 0 255, 2246 2509 >> [13, 1; 193, 1] >> 0 112, 2397 2509 >> [13, 1; 193, 1] >> 0 853, 1656 2521 >> Mat([2521, 1]) >> 1 71, 2450 2525 >> [5, 2; 101, 1] >> 0 293, 2232 2525 >> [5, 2; 101, 1] >> 0 818, 1707 2533 >> [17, 1; 149, 1] >> 0 701, 1832 2533 >> [17, 1; 149, 1] >> 0 999, 1534 2545 >> [5, 1; 509, 1] >> 0 208, 2337 2545 >> [5, 1; 509, 1] >> 0 717, 1828 2549 >> Mat([2549, 1]) >> 1 357, 2192 2557 >> Mat([2557, 1]) >> 1 611, 1946 2561 >> [13, 1; 197, 1] >> 0 408, 2153 2561 >> [13, 1; 197, 1] >> 0 577, 1984 2581 >> [29, 1; 89, 1] >> 0 568, 2013 2581 >> [29, 1; 89, 1] >> 0 945, 1636 2593 >> Mat([2593, 1]) >> 1 918, 1675 2605 >> [5, 1; 521, 1] >> 0 807, 1798 2605 >> [5, 1; 521, 1] >> 0 1277, 1328 2609 >> Mat([2609, 1]) >> 1 389, 2220 2617 >> Mat([2617, 1]) >> 1 667, 1950 2621 >> Mat([2621, 1]) >> 1 472, 2149 2633 >> Mat([2633, 1]) >> 1 1224, 1409 2657 >> Mat([2657, 1]) >> 1 163, 2494 2665 >> [5, 1; 13, 1; 41, 1] >> 0 73, 2592 2665 >> [5, 1; 13, 1; 41, 1] >> 0 278, 2387 2665 >> [5, 1; 13, 1; 41, 1] >> 0 788, 1877 2665 >> [5, 1; 13, 1; 41, 1] >> 0 993, 1672 2669 >> [17, 1; 157, 1] >> 0 914, 1755 2669 >> [17, 1; 157, 1] >> 0 1228, 1441 2677 >> Mat([2677, 1]) >> 1 550, 2127 2689 >> Mat([2689, 1]) >> 1 1142, 1547 2693 >> Mat([2693, 1]) >> 1 859, 1834 2701 >> [37, 1; 73, 1] >> 0 265, 2436 2701 >> [37, 1; 73, 1] >> 0 1141, 1560 2705 >> [5, 1; 541, 1] >> 0 52, 2653 2705 >> [5, 1; 541, 1] >> 0 593, 2112 2713 >> Mat([2713, 1]) >> 1 887, 1826 2725 >> [5, 2; 109, 1] >> 0 1057, 1668 2725 >> [5, 2; 109, 1] >> 0 1232, 1493 2729 >> Mat([2729, 1]) >> 1 1102, 1627 2741 >> Mat([2741, 1]) >> 1 656, 2085 2749 >> Mat([2749, 1]) >> 1 640, 2109 2753 >> Mat([2753, 1]) >> 1 794, 1959 2777 >> Mat([2777, 1]) >> 1 190, 2587 2785 >> [5, 1; 557, 1] >> 0 118, 2667 2785 >> [5, 1; 557, 1] >> 0 1232, 1553 2789 >> Mat([2789, 1]) >> 1 167, 2622 2797 >> Mat([2797, 1]) >> 1 603, 2194 2801 >> Mat([2801, 1]) >> 1 1258, 1543 2809 >> Mat([53, 2]) >> 0 500, 2309 2813 >> [29, 1; 97, 1] >> 0 75, 2738 2813 >> [29, 1; 97, 1] >> 0 1380, 1433 2825 >> [5, 2; 113, 1] >> 0 693, 2132 2825 >> [5, 2; 113, 1] >> 0 1032, 1793 2833 >> Mat([2833, 1]) >> 1 1357, 1476 2837 >> Mat([2837, 1]) >> 1 416, 2421 2845 >> [5, 1; 569, 1] >> 0 483, 2362 2845 >> [5, 1; 569, 1] >> 0 1052, 1793 2857 >> Mat([2857, 1]) >> 1 896, 1961 2861 >> Mat([2861, 1]) >> 1 1202, 1659 2873 >> [13, 2; 17, 1] >> 0 268, 2605 2873 >> [13, 2; 17, 1] >> 0 1084, 1789 2885 >> [5, 1; 577, 1] >> 0 553, 2332 2885 >> [5, 1; 577, 1] >> 0 1178, 1707 2897 >> Mat([2897, 1]) >> 1 1120, 1777 2909 >> Mat([2909, 1]) >> 1 878, 2031 2917 >> Mat([2917, 1]) >> 1 54, 2863 2929 >> [29, 1; 101, 1] >> 0 394, 2535 2929 >> [29, 1; 101, 1] >> 0 1404, 1525 2941 >> [17, 1; 173, 1] >> 0 599, 2342 2941 >> [17, 1; 173, 1] >> 0 1118, 1823 2953 >> Mat([2953, 1]) >> 1 1226, 1727 2957 >> Mat([2957, 1]) >> 1 1222, 1735 2965 >> [5, 1; 593, 1] >> 0 77, 2888 2965 >> [5, 1; 593, 1] >> 0 1263, 1702 2969 >> Mat([2969, 1]) >> 1 964, 2005 2977 >> [13, 1; 229, 1] >> 0 122, 2855 2977 >> [13, 1; 229, 1] >> 0 580, 2397 2993 >> [41, 1; 73, 1] >> 0 173, 2820 2993 >> [41, 1; 73, 1] >> 0 319, 2674 3001 >> Mat([3001, 1]) >> 1 1353, 1648 3005 >> [5, 1; 601, 1] >> 0 1077, 1928 3005 >> [5, 1; 601, 1] >> 0 1327, 1678 3029 >> [13, 1; 233, 1] >> 0 788, 2241 3029 >> [13, 1; 233, 1] >> 0 1487, 1542 3037 >> Mat([3037, 1]) >> 1 281, 2756 3041 >> Mat([3041, 1]) >> 1 774, 2267 3049 >> Mat([3049, 1]) >> 1 475, 2574 3061 >> Mat([3061, 1]) >> 1 501, 2560 3065 >> [5, 1; 613, 1] >> 0 578, 2487 3065 >> [5, 1; 613, 1] >> 0 648, 2417 3077 >> [17, 1; 181, 1] >> 0 200, 2877 3077 >> [17, 1; 181, 1] >> 0 1067, 2010 3085 >> [5, 1; 617, 1] >> 0 423, 2662 3085 >> [5, 1; 617, 1] >> 0 1428, 1657 3089 >> Mat([3089, 1]) >> 1 393, 2696 3109 >> Mat([3109, 1]) >> 1 727, 2382 3121 >> Mat([3121, 1]) >> 1 79, 3042 3125 >> Mat([5, 5]) >> 0 1068, 2057 3133 >> [13, 1; 241, 1] >> 0 177, 2956 3133 >> [13, 1; 241, 1] >> 0 1269, 1864 3137 >> Mat([3137, 1]) >> 1 56, 3081 3145 >> [5, 1; 17, 1; 37, 1] >> 0 302, 2843 3145 >> [5, 1; 17, 1; 37, 1] >> 0 327, 2818 3145 >> [5, 1; 17, 1; 37, 1] >> 0 438, 2707 3145 >> [5, 1; 17, 1; 37, 1] >> 0 1067, 2078 3161 >> [29, 1; 109, 1] >> 0 360, 2801 3161 >> [29, 1; 109, 1] >> 0 621, 2540 3169 >> Mat([3169, 1]) >> 1 1325, 1844 3181 >> Mat([3181, 1]) >> 1 282, 2899 3205 >> [5, 1; 641, 1] >> 0 487, 2718 3205 >> [5, 1; 641, 1] >> 0 1128, 2077 3209 >> Mat([3209, 1]) >> 1 484, 2725 3217 >> Mat([3217, 1]) >> 1 1436, 1781 3221 >> Mat([3221, 1]) >> 1 234, 2987 3229 >> Mat([3229, 1]) >> 1 839, 2390 3233 >> [53, 1; 61, 1] >> 0 560, 2673 3233 >> [53, 1; 61, 1] >> 0 1514, 1719 3253 >> Mat([3253, 1]) >> 1 1598, 1655 3257 >> Mat([3257, 1]) >> 1 291, 2966 3265 >> [5, 1; 653, 1] >> 0 802, 2463 3265 >> [5, 1; 653, 1] >> 0 1157, 2108 3277 >> [29, 1; 113, 1] >> 0 128, 3149 3277 >> [29, 1; 113, 1] >> 0 1032, 2245 3281 >> [17, 1; 193, 1] >> 0 81, 3200 3281 >> [17, 1; 193, 1] >> 0 1432, 1849 3293 >> [37, 1; 89, 1] >> 0 746, 2547 3293 >> [37, 1; 89, 1] >> 0 1301, 1992 3301 >> Mat([3301, 1]) >> 1 1212, 2089 3305 >> [5, 1; 661, 1] >> 0 767, 2538 3305 >> [5, 1; 661, 1] >> 0 1428, 1877 3313 >> Mat([3313, 1]) >> 1 407, 2906 3329 >> Mat([3329, 1]) >> 1 1600, 1729 3341 >> [13, 1; 257, 1] >> 0 1269, 2072 3341 >> [13, 1; 257, 1] >> 0 1526, 1815 3349 >> [17, 1; 197, 1] >> 0 183, 3166 3349 >> [17, 1; 197, 1] >> 0 999, 2350 3361 >> Mat([3361, 1]) >> 1 900, 2461 3365 >> [5, 1; 673, 1] >> 0 58, 3307 3365 >> [5, 1; 673, 1] >> 0 1288, 2077 3373 >> Mat([3373, 1]) >> 1 1105, 2268 3385 >> [5, 1; 677, 1] >> 0 703, 2682 3385 >> [5, 1; 677, 1] >> 0 1328, 2057 3389 >> Mat([3389, 1]) >> 1 1344, 2045 3413 >> Mat([3413, 1]) >> 1 1471, 1942 3425 >> [5, 2; 137, 1] >> 0 1407, 2018 3425 >> [5, 2; 137, 1] >> 0 1607, 1818 3433 >> Mat([3433, 1]) >> 1 1651, 1782 3445 >> [5, 1; 13, 1; 53, 1] >> 0 83, 3362 3445 >> [5, 1; 13, 1; 53, 1] >> 0 242, 3203 3445 >> [5, 1; 13, 1; 53, 1] >> 0 447, 2998 3445 >> [5, 1; 13, 1; 53, 1] >> 0 772, 2673 3449 >> Mat([3449, 1]) >> 1 1122, 2327 3457 >> Mat([3457, 1]) >> 1 708, 2749 3461 >> Mat([3461, 1]) >> 1 1453, 2008 3469 >> Mat([3469, 1]) >> 1 1003, 2466 3485 >> [5, 1; 17, 1; 41, 1] >> 0 132, 3353 3485 >> [5, 1; 17, 1; 41, 1] >> 0 378, 3107 3485 >> [5, 1; 17, 1; 41, 1] >> 0 1262, 2223 3485 >> [5, 1; 17, 1; 41, 1] >> 0 1713, 1772 3497 >> [13, 1; 269, 1] >> 0 187, 3310 3497 >> [13, 1; 269, 1] >> 0 889, 2608 3505 >> [5, 1; 701, 1] >> 0 1267, 2238 3505 >> [5, 1; 701, 1] >> 0 1537, 1968 3517 >> Mat([3517, 1]) >> 1 596, 2921 3529 >> Mat([3529, 1]) >> 1 808, 2721 3533 >> Mat([3533, 1]) >> 1 548, 2985 3541 >> Mat([3541, 1]) >> 1 852, 2689 3545 >> [5, 1; 709, 1] >> 0 613, 2932 3545 >> [5, 1; 709, 1] >> 0 1322, 2223 3557 >> Mat([3557, 1]) >> 1 943, 2614 3581 >> Mat([3581, 1]) >> 1 364, 3217 3589 >> [37, 1; 97, 1] >> 0 216, 3373 3589 >> [37, 1; 97, 1] >> 0 1671, 1918 3593 >> Mat([3593, 1]) >> 1 1153, 2440 3601 >> [13, 1; 277, 1] >> 0 60, 3541 3601 >> [13, 1; 277, 1] >> 0 1048, 2553 3613 >> Mat([3613, 1]) >> 1 85, 3528 3617 >> Mat([3617, 1]) >> 1 1234, 2383 3625 >> [5, 3; 29, 1] >> 0 307, 3318 3625 >> [5, 3; 29, 1] >> 0 568, 3057 3637 >> Mat([3637, 1]) >> 1 1027, 2610 3649 >> [41, 1; 89, 1] >> 0 1034, 2615 3649 >> [41, 1; 89, 1] >> 0 1280, 2369 3653 >> [13, 1; 281, 1] >> 0 1071, 2582 3653 >> [13, 1; 281, 1] >> 0 1633, 2020 3665 >> [5, 1; 733, 1] >> 0 353, 3312 3665 >> [5, 1; 733, 1] >> 0 1113, 2552 3673 >> Mat([3673, 1]) >> 1 994, 2679 3677 >> Mat([3677, 1]) >> 1 1309, 2368 3697 >> Mat([3697, 1]) >> 1 1131, 2566 3701 >> Mat([3701, 1]) >> 1 1279, 2422 3709 >> Mat([3709, 1]) >> 1 1609, 2100 3721 >> Mat([61, 2]) >> 0 682, 3039 3725 >> [5, 2; 149, 1] >> 0 193, 3532 3725 >> [5, 2; 149, 1] >> 0 1832, 1893 3733 >> Mat([3733, 1]) >> 1 851, 2882 3737 >> [37, 1; 101, 1] >> 0 697, 3040 3737 >> [37, 1; 101, 1] >> 0 919, 2818 3757 >> [13, 1; 17, 2] >> 0 616, 3141 3757 >> [13, 1; 17, 2] >> 0 905, 2852 3761 >> Mat([3761, 1]) >> 1 604, 3157 3769 >> Mat([3769, 1]) >> 1 1445, 2324 3785 >> [5, 1; 757, 1] >> 0 87, 3698 3785 >> [5, 1; 757, 1] >> 0 1427, 2358 3793 >> Mat([3793, 1]) >> 1 803, 2990 3797 >> Mat([3797, 1]) >> 1 742, 3055 3805 >> [5, 1; 761, 1] >> 0 722, 3083 3805 >> [5, 1; 761, 1] >> 0 1483, 2322 3809 >> [13, 1; 293, 1] >> 0 138, 3671 3809 >> [13, 1; 293, 1] >> 0 1620, 2189 3821 >> Mat([3821, 1]) >> 1 376, 3445 3833 >> Mat([3833, 1]) >> 1 361, 3472 3845 >> [5, 1; 769, 1] >> 0 62, 3783 3845 >> [5, 1; 769, 1] >> 0 707, 3138 3853 >> Mat([3853, 1]) >> 1 1305, 2548 3865 >> [5, 1; 773, 1] >> 0 317, 3548 3865 >> [5, 1; 773, 1] >> 0 1863, 2002 3869 >> [53, 1; 73, 1] >> 0 1560, 2309 3869 >> [53, 1; 73, 1] >> 0 1779, 2090 3877 >> Mat([3877, 1]) >> 1 502, 3375 3881 >> Mat([3881, 1]) >> 1 197, 3684 3889 >> Mat([3889, 1]) >> 1 454, 3435 3893 >> [17, 1; 229, 1] >> 0 336, 3557 3893 >> [17, 1; 229, 1] >> 0 565, 3328 3917 >> Mat([3917, 1]) >> 1 835, 3082 3925 >> [5, 2; 157, 1] >> 0 443, 3482 3925 >> [5, 2; 157, 1] >> 0 757, 3168 3929 >> Mat([3929, 1]) >> 1 226, 3703 3961 >> [17, 1; 233, 1] >> 0 89, 3872 3961 >> [17, 1; 233, 1] >> 0 1254, 2707 3965 >> [5, 1; 13, 1; 61, 1] >> 0 538, 3427 3965 >> [5, 1; 13, 1; 61, 1] >> 0 1048, 2917 3965 >> [5, 1; 13, 1; 61, 1] >> 0 1087, 2878 3965 >> [5, 1; 13, 1; 61, 1] >> 0 1292, 2673 3973 >> [29, 1; 137, 1] >> 0 1607, 2366 3973 >> [29, 1; 137, 1] >> 0 1955, 2018 3977 >> [41, 1; 97, 1] >> 0 1239, 2738 3977 >> [41, 1; 97, 1] >> 0 1918, 2059 3985 >> [5, 1; 797, 1] >> 0 582, 3403 3985 >> [5, 1; 797, 1] >> 0 1012, 2973 3989 >> Mat([3989, 1]) >> 1 481, 3508 4001 >> Mat([4001, 1]) >> 1 899, 3102 4013 >> Mat([4013, 1]) >> 1 1230, 2783 4021 >> Mat([4021, 1]) >> 1 723, 3298 4033 >> [37, 1; 109, 1] >> 0 142, 3891 4033 >> [37, 1; 109, 1] >> 0 512, 3521 4045 >> [5, 1; 809, 1] >> 0 318, 3727 4045 >> [5, 1; 809, 1] >> 0 1127, 2918 4049 >> Mat([4049, 1]) >> 1 884, 3165 4057 >> Mat([4057, 1]) >> 1 1857, 2200 4069 >> [13, 1; 313, 1] >> 0 1227, 2842 4069 >> [13, 1; 313, 1] >> 0 1903, 2166 4073 >> Mat([4073, 1]) >> 1 549, 3524 4093 >> Mat([4093, 1]) >> 1 1059, 3034 4097 >> [17, 1; 241, 1] >> 0 64, 4033 4097 >> [17, 1; 241, 1] >> 0 659, 3438 4105 >> [5, 1; 821, 1] >> 0 1347, 2758 4105 >> [5, 1; 821, 1] >> 0 1937, 2168 4121 >> [13, 1; 317, 1] >> 0 203, 3918 4121 >> [13, 1; 317, 1] >> 0 837, 3284 4129 >> Mat([4129, 1]) >> 1 895, 3234 4133 >> Mat([4133, 1]) >> 1 733, 3400 4141 >> [41, 1; 101, 1] >> 0 91, 4050 4141 >> [41, 1; 101, 1] >> 0 1303, 2838 4145 >> [5, 1; 829, 1] >> 0 583, 3562 4145 >> [5, 1; 829, 1] >> 0 1412, 2733 4153 >> Mat([4153, 1]) >> 1 1643, 2510 4157 >> Mat([4157, 1]) >> 1 1761, 2396 4177 >> Mat([4177, 1]) >> 1 457, 3720 4181 >> [37, 1; 113, 1] >> 0 919, 3262 4181 >> [37, 1; 113, 1] >> 0 1597, 2584 4201 >> Mat([4201, 1]) >> 1 1154, 3047 4205 >> [5, 1; 29, 2] >> 0 882, 3323 4205 >> [5, 1; 29, 2] >> 0 1723, 2482 4217 >> Mat([4217, 1]) >> 1 1911, 2306 4225 >> [5, 2; 13, 2] >> 0 268, 3957 4225 >> [5, 2; 13, 2] >> 0 1282, 2943 4229 >> Mat([4229, 1]) >> 1 2082, 2147 4241 >> Mat([4241, 1]) >> 1 1044, 3197 4253 >> Mat([4253, 1]) >> 1 561, 3692 4261 >> Mat([4261, 1]) >> 1 721, 3540 4265 >> [5, 1; 853, 1] >> 0 333, 3932 4265 >> [5, 1; 853, 1] >> 0 1373, 2892 4273 >> Mat([4273, 1]) >> 1 1200, 3073 4285 >> [5, 1; 857, 1] >> 0 207, 4078 4285 >> [5, 1; 857, 1] >> 0 1507, 2778 4289 >> Mat([4289, 1]) >> 1 528, 3761 4297 >> Mat([4297, 1]) >> 1 1972, 2325 4321 >> [29, 1; 149, 1] >> 0 1148, 3173 4321 >> [29, 1; 149, 1] >> 0 2042, 2279 4325 >> [5, 2; 173, 1] >> 0 93, 4232 4325 >> [5, 2; 173, 1] >> 0 1118, 3207 4337 >> Mat([4337, 1]) >> 1 886, 3451 4349 >> Mat([4349, 1]) >> 1 608, 3741 4357 >> Mat([4357, 1]) >> 1 66, 4291 4369 >> [17, 1; 257, 1] >> 0 1526, 2843 4369 >> [17, 1; 257, 1] >> 0 1815, 2554 4373 >> Mat([4373, 1]) >> 1 1904, 2469 4381 >> [13, 1; 337, 1] >> 0 148, 4233 4381 >> [13, 1; 337, 1] >> 0 863, 3518 4385 >> [5, 1; 877, 1] >> 0 1028, 3357 4385 >> [5, 1; 877, 1] >> 0 1603, 2782 4397 >> Mat([4397, 1]) >> 1 505, 3892 4405 >> [5, 1; 881, 1] >> 0 387, 4018 4405 >> [5, 1; 881, 1] >> 0 1268, 3137 4409 >> Mat([4409, 1]) >> 1 332, 4077 4421 >> Mat([4421, 1]) >> 1 952, 3469 4441 >> Mat([4441, 1]) >> 1 2146, 2295 4453 >> [61, 1; 73, 1] >> 0 538, 3915 4453 >> [61, 1; 73, 1] >> 0 1414, 3039 4457 >> Mat([4457, 1]) >> 1 1880, 2577 4469 >> [41, 1; 109, 1] >> 0 360, 4109 4469 >> [41, 1; 109, 1] >> 0 1057, 3412 4481 >> Mat([4481, 1]) >> 1 276, 4205 4493 >> Mat([4493, 1]) >> 1 2213, 2280 4505 >> [5, 1; 17, 1; 53, 1] >> 0 242, 4263 4505 >> [5, 1; 17, 1; 53, 1] >> 0 1143, 3362 4505 >> [5, 1; 17, 1; 53, 1] >> 0 1772, 2733 4505 >> [5, 1; 17, 1; 53, 1] >> 0 1832, 2673 4513 >> Mat([4513, 1]) >> 1 95, 4418 4517 >> Mat([4517, 1]) >> 1 1474, 3043 4525 >> [5, 2; 181, 1] >> 0 343, 4182 4525 >> [5, 2; 181, 1] >> 0 743, 3782 4537 >> [13, 1; 349, 1] >> 0 213, 4324 4537 >> [13, 1; 349, 1] >> 0 1958, 2579 4549 >> Mat([4549, 1]) >> 1 1260, 3289 4553 >> [29, 1; 157, 1] >> 0 1699, 2854 4553 >> [29, 1; 157, 1] >> 0 2013, 2540 4561 >> Mat([4561, 1]) >> 1 2205, 2356 4573 >> [17, 1; 269, 1] >> 0 1696, 2877 4573 >> [17, 1; 269, 1] >> 0 2070, 2503 4589 >> [13, 1; 353, 1] >> 0 395, 4194 4589 >> [13, 1; 353, 1] >> 0 1370, 3219 4597 >> Mat([4597, 1]) >> 1 2129, 2468 4621 >> Mat([4621, 1]) >> 1 152, 4469 4625 >> [5, 3; 37, 1] >> 0 68, 4557 4625 >> [5, 3; 37, 1] >> 0 1807, 2818 4633 >> [41, 1; 113, 1] >> 0 1567, 3066 4633 >> [41, 1; 113, 1] >> 0 1936, 2697 4637 >> Mat([4637, 1]) >> 1 2044, 2593 4645 >> [5, 1; 929, 1] >> 0 1253, 3392 4645 >> [5, 1; 929, 1] >> 0 2182, 2463 4649 >> Mat([4649, 1]) >> 1 1846, 2803 4657 >> Mat([4657, 1]) >> 1 1912, 2745 4673 >> Mat([4673, 1]) >> 1 1993, 2680 4685 >> [5, 1; 937, 1] >> 0 1133, 3552 4685 >> [5, 1; 937, 1] >> 0 1678, 3007 4705 >> [5, 1; 941, 1] >> 0 97, 4608 4705 >> [5, 1; 941, 1] >> 0 1038, 3667 4709 >> [17, 1; 277, 1] >> 0 217, 4492 4709 >> [17, 1; 277, 1] >> 0 1602, 3107 4717 >> [53, 1; 89, 1] >> 0 500, 4217 4717 >> [53, 1; 89, 1] >> 0 924, 3793 4721 >> Mat([4721, 1]) >> 1 1697, 3024 4729 >> Mat([4729, 1]) >> 1 1365, 3364 4733 >> Mat([4733, 1]) >> 1 897, 3836 4745 >> [5, 1; 13, 1; 73, 1] >> 0 538, 4207 4745 >> [5, 1; 13, 1; 73, 1] >> 0 1487, 3258 4745 >> [5, 1; 13, 1; 73, 1] >> 0 1633, 3112 4745 >> [5, 1; 13, 1; 73, 1] >> 0 2163, 2582 4765 >> [5, 1; 953, 1] >> 0 442, 4323 4765 >> [5, 1; 953, 1] >> 0 2348, 2417 4777 >> [17, 1; 281, 1] >> 0 1177, 3600 4777 >> [17, 1; 281, 1] >> 0 1458, 3319 4789 >> Mat([4789, 1]) >> 1 1481, 3308 4793 >> Mat([4793, 1]) >> 1 1480, 3313 4801 >> Mat([4801, 1]) >> 1 1403, 3398 4813 >> Mat([4813, 1]) >> 1 1868, 2945 4817 >> Mat([4817, 1]) >> 1 1291, 3526 4825 >> [5, 2; 193, 1] >> 0 1432, 3393 4825 >> [5, 2; 193, 1] >> 0 1818, 3007 4849 >> [13, 1; 373, 1] >> 0 642, 4207 4849 >> [13, 1; 373, 1] >> 0 850, 3999 4861 >> Mat([4861, 1]) >> 1 493, 4368 4877 >> Mat([4877, 1]) >> 1 719, 4158 4885 >> [5, 1; 977, 1] >> 0 252, 4633 4885 >> [5, 1; 977, 1] >> 0 1702, 3183 4889 >> Mat([4889, 1]) >> 1 730, 4159 4901 >> [13, 2; 29, 1] >> 0 70, 4831 4901 >> [13, 2; 29, 1] >> 0 99, 4802 4909 >> Mat([4909, 1]) >> 1 1613, 3296 4913 >> Mat([17, 3]) >> 0 1985, 2928 4925 >> [5, 2; 197, 1] >> 0 1168, 3757 4925 >> [5, 2; 197, 1] >> 0 1393, 3532 4933 >> Mat([4933, 1]) >> 1 1194, 3739 4937 >> Mat([4937, 1]) >> 1 849, 4088 4957 >> Mat([4957, 1]) >> 1 359, 4598 4969 >> Mat([4969, 1]) >> 1 1076, 3893 4973 >> Mat([4973, 1]) >> 1 223, 4750 4981 >> [17, 1; 293, 1] >> 0 2189, 2792 4981 >> [17, 1; 293, 1] >> 0 2206, 2775 4985 >> [5, 1; 997, 1] >> 0 1158, 3827 4985 >> [5, 1; 997, 1] >> 0 1833, 3152 4993 >> Mat([4993, 1]) >> 1 158, 4835 5009 >> Mat([5009, 1]) >> 1 539, 4470 5017 >> [29, 1; 173, 1] >> 0 945, 4072 5017 >> [29, 1; 173, 1] >> 0 1810, 3207 5021 >> Mat([5021, 1]) >> 1 1363, 3658 5045 >> [5, 1; 1009, 1] >> 0 1478, 3567 5045 >> [5, 1; 1009, 1] >> 0 2487, 2558 5057 >> [13, 1; 389, 1] >> 0 1282, 3775 5057 >> [13, 1; 389, 1] >> 0 2449, 2608 5065 >> [5, 1; 1013, 1] >> 0 968, 4097 5065 >> [5, 1; 1013, 1] >> 0 1058, 4007 5069 >> [37, 1; 137, 1] >> 0 1881, 3188 5069 >> [37, 1; 137, 1] >> 0 1955, 3114 5077 >> Mat([5077, 1]) >> 1 858, 4219 5081 >> Mat([5081, 1]) >> 1 2412, 2669 5101 >> Mat([5101, 1]) >> 1 101, 5000 5105 >> [5, 1; 1021, 1] >> 0 647, 4458 5105 >> [5, 1; 1021, 1] >> 0 1668, 3437 5113 >> Mat([5113, 1]) >> 1 2025, 3088 5125 >> [5, 3; 41, 1] >> 0 1057, 4068 5125 >> [5, 3; 41, 1] >> 0 2182, 2943 5141 >> [53, 1; 97, 1] >> 0 507, 4634 5141 >> [53, 1; 97, 1] >> 0 560, 4581 5153 >> Mat([5153, 1]) >> 1 227, 4926 5161 >> [13, 1; 397, 1] >> 0 460, 4701 5161 >> [13, 1; 397, 1] >> 0 2319, 2842 5165 >> [5, 1; 1033, 1] >> 0 678, 4487 5165 >> [5, 1; 1033, 1] >> 0 1388, 3777 5185 >> [5, 1; 17, 1; 61, 1] >> 0 72, 5113 5185 >> [5, 1; 17, 1; 61, 1] >> 0 438, 4747 5185 >> [5, 1; 17, 1; 61, 1] >> 0 2002, 3183 5185 >> [5, 1; 17, 1; 61, 1] >> 0 2512, 2673 5189 >> Mat([5189, 1]) >> 1 2446, 2743 5197 >> Mat([5197, 1]) >> 1 1969, 3228 5209 >> Mat([5209, 1]) >> 1 2098, 3111 5213 >> [13, 1; 401, 1] >> 0 421, 4792 5213 >> [13, 1; 401, 1] >> 0 2426, 2787 5233 >> Mat([5233, 1]) >> 1 2253, 2980 5237 >> Mat([5237, 1]) >> 1 369, 4868 5245 >> [5, 1; 1049, 1] >> 0 623, 4622 5245 >> [5, 1; 1049, 1] >> 0 1672, 3573 5249 >> [29, 1; 181, 1] >> 0 162, 5087 5249 >> [29, 1; 181, 1] >> 0 1467, 3782 5261 >> Mat([5261, 1]) >> 1 827, 4434 5273 >> Mat([5273, 1]) >> 1 944, 4329 5281 >> Mat([5281, 1]) >> 1 1673, 3608 5297 >> Mat([5297, 1]) >> 1 2313, 2984 5305 >> [5, 1; 1061, 1] >> 0 103, 5202 5305 >> [5, 1; 1061, 1] >> 0 958, 4347 5309 >> Mat([5309, 1]) >> 1 1804, 3505 5317 >> [13, 1; 409, 1] >> 0 1084, 4233 5317 >> [13, 1; 409, 1] >> 0 1370, 3947 5321 >> [17, 1; 313, 1] >> 0 914, 4407 5321 >> [17, 1; 313, 1] >> 0 2529, 2792 5329 >> Mat([73, 2]) >> 0 776, 4553 5333 >> Mat([5333, 1]) >> 1 2630, 2703 5345 >> [5, 1; 1069, 1] >> 0 1318, 4027 5345 >> [5, 1; 1069, 1] >> 0 2387, 2958 5353 >> [53, 1; 101, 1] >> 0 394, 4959 5353 >> [53, 1; 101, 1] >> 0 818, 4535 5365 >> [5, 1; 29, 1; 37, 1] >> 0 302, 5063 5365 >> [5, 1; 29, 1; 37, 1] >> 0 882, 4483 5365 >> [5, 1; 29, 1; 37, 1] >> 0 2337, 3028 5365 >> [5, 1; 29, 1; 37, 1] >> 0 2448, 2917 5381 >> Mat([5381, 1]) >> 1 1739, 3642 5389 >> [17, 1; 317, 1] >> 0 837, 4552 5389 >> [17, 1; 317, 1] >> 0 2333, 3056 5393 >> Mat([5393, 1]) >> 1 665, 4728 5413 >> Mat([5413, 1]) >> 1 429, 4984 5417 >> Mat([5417, 1]) >> 1 368, 5049 5429 >> [61, 1; 89, 1] >> 0 233, 5196 5429 >> [61, 1; 89, 1] >> 0 1636, 3793 5437 >> Mat([5437, 1]) >> 1 630, 4807 5441 >> Mat([5441, 1]) >> 1 2452, 2989 5449 >> Mat([5449, 1]) >> 1 635, 4814 5465 >> [5, 1; 1093, 1] >> 0 563, 4902 5465 >> [5, 1; 1093, 1] >> 0 1623, 3842 5473 >> [13, 1; 421, 1] >> 0 450, 5023 5473 >> [13, 1; 421, 1] >> 0 1292, 4181 5477 >> Mat([5477, 1]) >> 1 74, 5403 5485 >> [5, 1; 1097, 1] >> 0 1438, 4047 5485 >> [5, 1; 1097, 1] >> 0 1853, 3632 5501 >> Mat([5501, 1]) >> 1 1115, 4386 5513 >> [37, 1; 149, 1] >> 0 105, 5408 5513 >> [37, 1; 149, 1] >> 0 1893, 3620 5521 >> Mat([5521, 1]) >> 1 765, 4756 5525 >> [5, 2; 13, 1; 17, 1] >> 0 268, 5257 5525 >> [5, 2; 13, 1; 17, 1] >> 0 1568, 3957 5525 >> [5, 2; 13, 1; 17, 1] >> 0 1968, 3557 5525 >> [5, 2; 13, 1; 17, 1] >> 0 2257, 3268 5545 >> [5, 1; 1109, 1] >> 0 1463, 4082 5545 >> [5, 1; 1109, 1] >> 0 2572, 2973 5557 >> Mat([5557, 1]) >> 1 2478, 3079 5569 >> Mat([5569, 1]) >> 1 973, 4596 5573 >> Mat([5573, 1]) >> 1 2017, 3556 5581 >> Mat([5581, 1]) >> 1 1437, 4144 5585 >> [5, 1; 1117, 1] >> 0 903, 4682 5585 >> [5, 1; 1117, 1] >> 0 2448, 3137 5597 >> [29, 1; 193, 1] >> 0 853, 4744 5597 >> [29, 1; 193, 1] >> 0 2042, 3555 5617 >> [41, 1; 137, 1] >> 0 237, 5380 5617 >> [41, 1; 137, 1] >> 0 2018, 3599 5629 >> [13, 1; 433, 1] >> 0 1045, 4584 5629 >> [13, 1; 433, 1] >> 0 2777, 2852 5641 >> Mat([5641, 1]) >> 1 1429, 4212 5645 >> [5, 1; 1129, 1] >> 0 168, 5477 5645 >> [5, 1; 1129, 1] >> 0 1297, 4348 5653 >> Mat([5653, 1]) >> 1 310, 5343 5657 >> Mat([5657, 1]) >> 1 1670, 3987 5669 >> Mat([5669, 1]) >> 1 1046, 4623 5689 >> Mat([5689, 1]) >> 1 2124, 3565 5693 >> Mat([5693, 1]) >> 1 1193, 4500 5701 >> Mat([5701, 1]) >> 1 385, 5316 5713 >> [29, 1; 197, 1] >> 0 1984, 3729 5713 >> [29, 1; 197, 1] >> 0 2772, 2941 5717 >> Mat([5717, 1]) >> 1 2416, 3301 5725 >> [5, 2; 229, 1] >> 0 107, 5618 5725 >> [5, 2; 229, 1] >> 0 2168, 3557 5729 >> [17, 1; 337, 1] >> 0 863, 4866 5729 >> [17, 1; 337, 1] >> 0 1874, 3855 5737 >> Mat([5737, 1]) >> 1 1126, 4611 5741 >> Mat([5741, 1]) >> 1 2378, 3363 5749 >> Mat([5749, 1]) >> 1 806, 4943 5765 >> [5, 1; 1153, 1] >> 0 1013, 4752 5765 >> [5, 1; 1153, 1] >> 0 1293, 4472 5777 >> [53, 1; 109, 1] >> 0 76, 5701 5777 >> [53, 1; 109, 1] >> 0 2256, 3521 5785 >> [5, 1; 13, 1; 89, 1] >> 0 1123, 4662 5785 >> [5, 1; 13, 1; 89, 1] >> 0 1568, 4217 5785 >> [5, 1; 13, 1; 89, 1] >> 0 1903, 3882 5785 >> [5, 1; 13, 1; 89, 1] >> 0 2348, 3437 5801 >> Mat([5801, 1]) >> 1 1145, 4656 5809 >> [37, 1; 157, 1] >> 0 1856, 3953 5809 >> [37, 1; 157, 1] >> 0 2226, 3583 5813 >> Mat([5813, 1]) >> 1 796, 5017 5821 >> Mat([5821, 1]) >> 1 1242, 4579 5825 >> [5, 2; 233, 1] >> 0 843, 4982 5825 >> [5, 2; 233, 1] >> 0 2707, 3118 5837 >> [13, 1; 449, 1] >> 0 382, 5455 5837 >> [13, 1; 449, 1] >> 0 2761, 3076 5849 >> Mat([5849, 1]) >> 1 2839, 3010 5857 >> Mat([5857, 1]) >> 1 1310, 4547 5861 >> Mat([5861, 1]) >> 1 754, 5107 5869 >> Mat([5869, 1]) >> 1 1042, 4827 5881 >> Mat([5881, 1]) >> 1 1098, 4783 5897 >> Mat([5897, 1]) >> 1 543, 5354 5905 >> [5, 1; 1181, 1] >> 0 243, 5662 5905 >> [5, 1; 1181, 1] >> 0 938, 4967 5917 >> [61, 1; 97, 1] >> 0 172, 5745 5917 >> [61, 1; 97, 1] >> 0 560, 5357 5933 >> [17, 1; 349, 1] >> 0 2656, 3277 5933 >> [17, 1; 349, 1] >> 0 2928, 3005 5941 >> [13, 1; 457, 1] >> 0 109, 5832 5941 >> [13, 1; 457, 1] >> 0 2176, 3765 5945 >> [5, 1; 29, 1; 41, 1] >> 0 278, 5667 5945 >> [5, 1; 29, 1; 41, 1] >> 0 1467, 4478 5945 >> [5, 1; 29, 1; 41, 1] >> 0 2018, 3927 5945 >> [5, 1; 29, 1; 41, 1] >> 0 2738, 3207 5953 >> Mat([5953, 1]) >> 1 2403, 3550 5965 >> [5, 1; 1193, 1] >> 0 1007, 4958 5965 >> [5, 1; 1193, 1] >> 0 2572, 3393 5981 >> Mat([5981, 1]) >> 1 1317, 4664 5989 >> [53, 1; 113, 1] >> 0 1454, 4535 5989 >> [53, 1; 113, 1] >> 0 1567, 4422 5993 >> [13, 1; 461, 1] >> 0 970, 5023 5993 >> [13, 1; 461, 1] >> 0 2257, 3736 6001 >> [17, 1; 353, 1] >> 0 395, 5606 6001 >> [17, 1; 353, 1] >> 0 1101, 4900 6005 >> [5, 1; 1201, 1] >> 0 1152, 4853 6005 >> [5, 1; 1201, 1] >> 0 2353, 3652 6025 >> [5, 2; 241, 1] >> 0 418, 5607 6025 >> [5, 2; 241, 1] >> 0 1382, 4643 6029 >> Mat([6029, 1]) >> 1 1801, 4228 6037 >> Mat([6037, 1]) >> 1 2652, 3385 6053 >> Mat([6053, 1]) >> 1 2832, 3221 6065 >> [5, 1; 1213, 1] >> 0 718, 5347 6065 >> [5, 1; 1213, 1] >> 0 1708, 4357 6073 >> Mat([6073, 1]) >> 1 2524, 3549 6085 >> [5, 1; 1217, 1] >> 0 78, 6007 6085 >> [5, 1; 1217, 1] >> 0 2512, 3573 6089 >> Mat([6089, 1]) >> 1 455, 5634 6101 >> Mat([6101, 1]) >> 1 247, 5854 6109 >> [41, 1; 149, 1] >> 0 1385, 4724 6109 >> [41, 1; 149, 1] >> 0 2428, 3681 6113 >> Mat([6113, 1]) >> 1 1089, 5024 6121 >> Mat([6121, 1]) >> 1 2583, 3538 6133 >> Mat([6133, 1]) >> 1 865, 5268 6145 >> [5, 1; 1229, 1] >> 0 597, 5548 6145 >> [5, 1; 1229, 1] >> 0 632, 5513 6161 >> [61, 1; 101, 1] >> 0 111, 6050 6161 >> [61, 1; 101, 1] >> 0 2939, 3222 6173 >> Mat([6173, 1]) >> 1 2447, 3726 6185 >> [5, 1; 1237, 1] >> 0 1783, 4402 6185 >> [5, 1; 1237, 1] >> 0 1928, 4257 6197 >> Mat([6197, 1]) >> 1 2007, 4190 6205 >> [5, 1; 17, 1; 73, 1] >> 0 557, 5648 6205 >> [5, 1; 17, 1; 73, 1] >> 0 922, 5283 6205 >> [5, 1; 17, 1; 73, 1] >> 0 1798, 4407 6205 >> [5, 1; 17, 1; 73, 1] >> 0 2163, 4042 6217 >> Mat([6217, 1]) >> 1 2372, 3845 6221 >> Mat([6221, 1]) >> 1 1121, 5100 6229 >> Mat([6229, 1]) >> 1 1451, 4778 6245 >> [5, 1; 1249, 1] >> 0 1913, 4332 6245 >> [5, 1; 1249, 1] >> 0 3083, 3162 6253 >> [13, 2; 37, 1] >> 0 746, 5507 6253 >> [13, 2; 37, 1] >> 0 2436, 3817 6257 >> Mat([6257, 1]) >> 1 1584, 4673 6269 >> Mat([6269, 1]) >> 1 1523, 4746 6277 >> Mat([6277, 1]) >> 1 1033, 5244 6301 >> Mat([6301, 1]) >> 1 2184, 4117 6305 >> [5, 1; 13, 1; 97, 1] >> 0 463, 5842 6305 >> [5, 1; 13, 1; 97, 1] >> 0 798, 5507 6305 >> [5, 1; 13, 1; 97, 1] >> 0 1477, 4828 6305 >> [5, 1; 13, 1; 97, 1] >> 0 2738, 3567 6317 >> Mat([6317, 1]) >> 1 1963, 4354 6329 >> Mat([6329, 1]) >> 1 2219, 4110 6337 >> Mat([6337, 1]) >> 1 178, 6159 6341 >> [17, 1; 373, 1] >> 0 642, 5699 6341 >> [17, 1; 373, 1] >> 0 2342, 3999 6353 >> Mat([6353, 1]) >> 1 1392, 4961 6361 >> Mat([6361, 1]) >> 1 1751, 4610 6373 >> Mat([6373, 1]) >> 1 1879, 4494 6385 >> [5, 1; 1277, 1] >> 0 113, 6272 6385 >> [5, 1; 1277, 1] >> 0 2667, 3718 6389 >> Mat([6389, 1]) >> 1 2092, 4297 6397 >> Mat([6397, 1]) >> 1 1302, 5095 6401 >> [37, 1; 173, 1] >> 0 80, 6321 6401 >> [37, 1; 173, 1] >> 0 253, 6148 6409 >> [13, 1; 17, 1; 29, 1] >> 0 684, 5725 6409 >> [13, 1; 17, 1; 29, 1] >> 0 1815, 4594 6409 >> [13, 1; 17, 1; 29, 1] >> 0 2163, 4246 6409 >> [13, 1; 17, 1; 29, 1] >> 0 3115, 3294 6421 >> Mat([6421, 1]) >> 1 825, 5596 6425 >> [5, 2; 257, 1] >> 0 2843, 3582 6425 >> [5, 2; 257, 1] >> 0 3068, 3357 6437 >> [41, 1; 157, 1] >> 0 1385, 5052 6437 >> [41, 1; 157, 1] >> 0 2697, 3740 6445 >> [5, 1; 1289, 1] >> 0 1768, 4677 6445 >> [5, 1; 1289, 1] >> 0 3057, 3388 6449 >> Mat([6449, 1]) >> 1 1854, 4595 6469 >> Mat([6469, 1]) >> 1 2977, 3492 6473 >> Mat([6473, 1]) >> 1 1808, 4665 6481 >> Mat([6481, 1]) >> 1 729, 5752 6485 >> [5, 1; 1297, 1] >> 0 1333, 5152 6485 >> [5, 1; 1297, 1] >> 0 2558, 3927 6497 >> [73, 1; 89, 1] >> 0 411, 6086 6497 >> [73, 1; 89, 1] >> 0 1725, 4772 6505 >> [5, 1; 1301, 1] >> 0 1352, 5153 6505 >> [5, 1; 1301, 1] >> 0 2653, 3852 6521 >> Mat([6521, 1]) >> 1 2364, 4157 6529 >> Mat([6529, 1]) >> 1 2311, 4218 6553 >> Mat([6553, 1]) >> 1 3186, 3367 6565 >> [5, 1; 13, 1; 101, 1] >> 0 697, 5868 6565 >> [5, 1; 13, 1; 101, 1] >> 0 798, 5767 6565 >> [5, 1; 13, 1; 101, 1] >> 0 1828, 4737 6565 >> [5, 1; 13, 1; 101, 1] >> 0 3242, 3323 6569 >> Mat([6569, 1]) >> 1 3038, 3531 6577 >> Mat([6577, 1]) >> 1 1624, 4953 6581 >> Mat([6581, 1]) >> 1 2727, 3854 6605 >> [5, 1; 1321, 1] >> 0 257, 6348 6605 >> [5, 1; 1321, 1] >> 0 1578, 5027 6613 >> [17, 1; 389, 1] >> 0 115, 6498 6613 >> [17, 1; 389, 1] >> 0 1441, 5172 6617 >> [13, 1; 509, 1] >> 0 1828, 4789 6617 >> [13, 1; 509, 1] >> 0 2244, 4373 6625 >> [5, 3; 53, 1] >> 0 182, 6443 6625 >> [5, 3; 53, 1] >> 0 818, 5807 6637 >> Mat([6637, 1]) >> 1 2828, 3809 6641 >> [29, 1; 229, 1] >> 0 336, 6305 6641 >> [29, 1; 229, 1] >> 0 1496, 5145 6649 >> [61, 1; 109, 1] >> 0 294, 6355 6649 >> [61, 1; 109, 1] >> 0 621, 6028 6653 >> Mat([6653, 1]) >> 1 752, 5901 6661 >> Mat([6661, 1]) >> 1 658, 6003 6673 >> Mat([6673, 1]) >> 1 2437, 4236 6689 >> Mat([6689, 1]) >> 1 2759, 3930 6697 >> [37, 1; 181, 1] >> 0 524, 6173 6697 >> [37, 1; 181, 1] >> 0 1067, 5630 6701 >> Mat([6701, 1]) >> 1 1721, 4980 6709 >> Mat([6709, 1]) >> 1 2150, 4559 6725 >> [5, 2; 269, 1] >> 0 82, 6643 6725 >> [5, 2; 269, 1] >> 0 1532, 5193 6733 >> Mat([6733, 1]) >> 1 2217, 4516 6737 >> Mat([6737, 1]) >> 1 2393, 4344 6749 >> [17, 1; 397, 1] >> 0 1254, 5495 6749 >> [17, 1; 397, 1] >> 0 2716, 4033 6757 >> [29, 1; 233, 1] >> 0 2419, 4338 6757 >> [29, 1; 233, 1] >> 0 3173, 3584 6761 >> Mat([6761, 1]) >> 1 1775, 4986 6773 >> [13, 1; 521, 1] >> 0 2319, 4454 6773 >> [13, 1; 521, 1] >> 0 2891, 3882 6781 >> Mat([6781, 1]) >> 1 995, 5786 6793 >> Mat([6793, 1]) >> 1 709, 6084 6805 >> [5, 1; 1361, 1] >> 0 747, 6058 6805 >> [5, 1; 1361, 1] >> 0 2108, 4697 6817 >> [17, 1; 401, 1] >> 0 421, 6396 6817 >> [17, 1; 401, 1] >> 0 1985, 4832 6829 >> Mat([6829, 1]) >> 1 1596, 5233 6833 >> Mat([6833, 1]) >> 1 1307, 5526 6841 >> Mat([6841, 1]) >> 1 1625, 5216 6845 >> [5, 1; 37, 2] >> 0 117, 6728 6845 >> [5, 1; 37, 2] >> 0 1252, 5593 6857 >> Mat([6857, 1]) >> 1 Exercise for the readers from the previous centuries: Come up with a routine to generate n's with complete set of known (a, b) pairs to determine if they have only-one or more prime factors. Last fiddled with by a1call on 2021-01-12 at 05:27
 2021-01-12, 09:32 #8 Nick     Dec 2012 The Netherlands 62E16 Posts Try Gaussian integers!
 2021-01-13, 07:53 #9 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7×283 Posts While the concept does not apply to Mersenne numbers since valuation(Mn,2)==1 It does apply to Fermat numbers greater than F0. The following code will find the 1st (a, b) pair for Fermat numbers greater than F0 virtually-instantly: Code: for(n=1,9,{ fermatNumber = 2^(2^n)+1; print("\nfermatNumber = ",fermatNumber ); a = sqrtint(fermatNumber -1);print("a = ",a ); b=fermatNumber -a;;print("b = ",b ); m=lift(Mod(a*b,fermatNumber );); print("F",n," >>-->> m = ",m); }) Output: Code: fermatNumber = 5 a = 2 b = 3 F1 >>-->> m = 1 fermatNumber = 17 a = 4 b = 13 F2 >>-->> m = 1 fermatNumber = 257 a = 16 b = 241 F3 >>-->> m = 1 fermatNumber = 65537 a = 256 b = 65281 F4 >>-->> m = 1 fermatNumber = 4294967297 a = 65536 b = 4294901761 F5 >>-->> m = 1 fermatNumber = 18446744073709551617 a = 4294967296 b = 18446744069414584321 F6 >>-->> m = 1 fermatNumber = 340282366920938463463374607431768211457 a = 18446744073709551616 b = 340282366920938463444927863358058659841 F7 >>-->> m = 1 fermatNumber = 115792089237316195423570985008687907853269984665640564039457584007913129639937 a = 340282366920938463463374607431768211456 b = 115792089237316195423570985008687907852929702298719625575994209400481361428481 F8 >>-->> m = 1 fermatNumber = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 a = 115792089237316195423570985008687907853269984665640564039457584007913129639936 b = 13407807929942597099574024998205846127479365820592393377723561443721764030073431184712636981971479856705023170278632780869088242247907112362425735876444161 F9 >>-->> m = 1 For F1 to F4 there are no other positive integer pairs (a, b) other than listed. For F5 however there is a 2nd pair: Code: \\DTC-120-A From Rashid Naimi - 1/13/2321 BC F5 = 2^(2^5)+1 a = 46837383 b = F5-a (a*b-1)/F5 Output: Code: (02:48) gp > F5 = 2^(2^5)+1 %31 = 4294967297 (02:48) gp > a = 46837383 %32 = 46837383 (02:48) gp > b = F5-a %33 = 4248129914 (02:48) gp > (a*b-1)/F5 %34 = 46326613 Unfortunately I have no clue how to find the secondary (a, b) pairs without Bruce-Lee Brute-Force. Thank you for your time.

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