/* ** search for Pisot sequences that almost satisfy a linear recurrence (arXiv:1609.05570v1) ** ** it looks like $x=10$, $y=19+10^2k$, $r=1/2$, gives rise to longer and longer fake linear ** recurrences, of the form a_n = (12+10k)a_{n-1} - (2+k)a_{n-2} + (10+8k)a_{n-3} - (6+5k)a_{n-4} ** this was tested for $0 \leq k < 100$. Does this pattern continue indefinitely? ** this phenomenon also appears for other values of $x$ and for $y$ in an arithmetic progression; ** for example, $x=23$ and $y=124+23^2k$, or $y=126+23^2k$, or $y=403+23^2k$, or $y=405+23^2k$ * also for $x=30$, $y=989+30^2k$? ** D. G. Cantor appears to have studies sequences of this form ** ** 25 longest fake sequences for r=0, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000022964456310 161 683 54068 13 [4,1,0,0,2,-1,0,0,0,-2,-2,0,-1] ** 1.000266808359533 481 1654 8279 14 [4,-2,0,1,0,-3,3,-2,0,1,1,-3,2,-1] ** 1.000091645618898 417 1400 7693 16 [3,0,4,0,1,-2,-1,-2,0,0,1,0,0,-1,-1,-1] ** 1.000142374097160 387 1296 6785 12 [3,1,1,-1,-1,-2,0,0,1,1,0,-2] ** 1.000245959654823 310 899 5790 18 [2,2,2,0,-1,-2,-2,-1,0,1,2,0,-1,-1,-1,0,0,1] ** 1.000268636529352 280 676 5629 16 [3,-2,2,-1,-1,0,0,-1,2,-1,1,-1,1,-1,1,-1] ** 1.000268636529352 676 1632 5628 16 [3,-2,2,-1,-1,0,0,-1,2,-1,1,-1,1,-1,1,-1] ** 1.000348262482604 41 276 5095 27 [7,-2,1,1,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1] ** 1.000348262482604 276 1857 5094 27 [7,-2,1,1,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1] ** 1.000416303546555 293 883 4633 13 [3,0,0,1,-2,0,1,-1,2,0,-2,0,-1] ** 1.000527961890823 383 1668 3054 11 [4,2,-2,0,0,2,-1,-2,1,-2,-1] ** 1.000479191568966 165 561 2602 10 [4,-2,0,-1,2,-1,0,0,1,-2] ** 1.000479191568966 561 1907 2601 10 [4,-2,0,-1,2,-1,0,0,1,-2] ** 1.001214184615256 464 1616 2560 18 [5,-7,8,-9,8,-5,3,-2,1,-1,1,-1,1,-1,1,-1,1,-1] ** 1.000470307092707 489 1097 2369 13 [3,-2,1,-1,1,-1,0,1,-1,0,0,1,-1] ** 1.000519113342433 377 988 1908 14 [3,-1,0,0,0,1,-2,0,0,0,0,-1,2,-1] ** 1.000460258220997 190 430 1862 21 [3,-1,-2,1,0,1,-1,-1,2,-1,-1,1,0,0,1,-1,0,0,0,1,-1] ** 1.000460258220997 430 973 1861 21 [3,-1,-2,1,0,1,-1,-1,2,-1,-1,1,0,0,1,-1,0,0,0,1,-1] ** 1.001009258596126 247 823 1795 16 [4,-3,4,-6,5,-4,5,-3,0,-1,-1,2,-1,1,-2,1] ** 1.000302834273007 306 1630 1730 10 [5,2,-1,-2,-1,2,2,-3,-1,-2] ** 1.000555891890885 241 700 1589 12 [3,0,-1,1,-1,-1,0,1,0,0,1,-2] ** 1.001437520701664 547 1974 1500 11 [3,2,1,-1,0,-1,0,1,0,-2,-2] ** 1.000316954940482 5 1977 1486 10 [395,158,0,-79,-79,-79,-79,-79,-79,-79] ** 1.000321024961523 5 1952 1467 10 [390,156,0,-78,-78,-78,-78,-78,-78,-78] ** 1.001597047141210 378 1501 1454 9 [4,-1,4,-2,1,-3,-1,0,-1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=1/5, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000035446251236 193 1893 10891 8 [11,-12,3,0,5,-5,3,-3] ** 1.000226247803807 372 832 8889 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1] ** 1.000226247803807 832 1861 8888 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1] ** 1.000101629707544 399 1831 8730 14 [4,2,3,1,0,0,0,-1,-2,-2,-1,-1,-1,-1] ** 1.000115254239615 114 825 8515 10 [7,2,-2,-1,2,1,1,0,1,-4] ** 1.000118136558231 207 1379 6740 6 [6,5,-4,0,2,-2] ** 1.000033867040633 79 1146 6661 9 [15,-7,-2,-6,4,6,-2,3,-9] ** 1.000146782468316 416 1264 6490 12 [3,0,1,-2,0,0,1,0,-2,-1,0,1] ** 1.000167938722114 297 1543 5979 12 [4,5,6,2,-2,-4,-5,-3,-3,0,1,1] ** 1.000194176476517 114 1726 5243 18 [15,2,2,-2,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,-2] ** 1.000152037699776 121 761 4790 6 [6,2,-1,-1,1,-4] ** 1.000193180960489 240 1510 4489 11 [7,-5,3,3,-3,2,-2,1,-1,1,-1] ** 1.000405435601516 269 744 4076 9 [3,0,-2,0,2,-1,-1,1,-1] ** 1.000189073335693 116 1504 3820 5 [12,12,7,-4,1] ** 1.000161465182546 431 1932 3801 8 [5,-3,4,-5,4,-4,3,-1] ** 1.000108265319972 103 1167 3647 4 [11,4,-2,-11] ** 1.000166937059998 263 1553 3456 13 [6,0,-3,-2,0,3,1,1,-2,-3,1,2,-2] ** 1.000279103859397 82 444 3145 8 [6,-3,-1,0,2,1,-2,-1] ** 1.000282757088752 110 495 2920 11 [4,3,-3,-2,1,0,-2,0,-1,0,2] ** 1.000463969727674 124 424 2662 7 [4,-3,4,-2,1,-1,-1] ** 1.000463969727674 424 1450 2661 7 [4,-3,4,-2,1,-1,-1] ** 1.000555891890885 241 700 2476 12 [3,0,-1,1,-1,-1,0,1,0,0,1,-2] ** 1.000467656763371 66 291 2291 14 [4,2,0,-4,0,2,2,-2,-2,0,4,0,-2,-3] ** 1.000467656763371 291 1283 2290 14 [4,2,0,-4,0,2,2,-2,-2,0,4,0,-2,-3] ** 1.000020550093545 303 947 2277 9 [3,0,1,1,-1,0,0,-1,-1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=1/4, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000089148691747 29 1070 26527 25 [37,-4,6,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,14] ** 1.000090412637074 29 1911 26137 25 [66,-7,11,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,25] ** 1.000035446251236 193 1893 17236 8 [11,-12,3,0,5,-5,3,-3] ** 1.000115254239615 114 825 10438 10 [7,2,-2,-1,2,1,1,0,1,-4] ** 1.000107273937310 386 1684 9345 8 [4,2,-2,1,-1,0,1,-2] ** 1.000226247803807 372 832 8827 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1] ** 1.000226247803807 832 1861 8826 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1] ** 1.000118136558231 207 1379 8660 6 [6,5,-4,0,2,-2] ** 1.000034775162274 113 729 7924 11 [7,-4,3,0,-1,-3,-1,1,-3,1,2] ** 1.000116397870165 351 1097 6691 15 [3,1,-2,0,1,0,-1,0,2,0,-2,1,1,-1,-1] ** 1.000152037699776 121 761 6247 6 [6,2,-1,-1,1,-4] ** 1.000166040411401 224 1358 6086 25 [6,0,2,2,-1,-1,1,0,-1,0,0,-1,-1,0,0,-1,0,0,-1,0,0,-1,0,0,-1] ** 1.000108265319972 103 1167 5704 4 [11,4,-2,-11] ** 1.000146782468316 416 1264 5616 12 [3,0,1,-2,0,0,1,0,-2,-1,0,1] ** 1.000107700453608 156 1108 5377 8 [8,-7,5,-5,7,-7,2,1] ** 1.000161465182546 431 1932 5177 8 [5,-3,4,-5,4,-4,3,-1] ** 1.000166937059998 263 1553 4766 13 [6,0,-3,-2,0,3,1,1,-2,-3,1,2,-2] ** 1.000188345303973 49 1889 4066 13 [39,-17,-12,7,3,6,-19,5,-11,-1,8,-5,11] ** 1.000279103859397 82 444 3930 8 [6,-3,-1,0,2,1,-2,-1] ** 1.000282757088752 110 495 3716 11 [4,3,-3,-2,1,0,-2,0,-1,0,2] ** 1.000214865252435 88 1816 3261 4 [21,-8,11,-19] ** 1.000186647603218 32 1943 3075 12 [61,-17,-6,19,-2,6,-19,2,-6,17,6,-19] ** 1.000243869198878 246 1921 2937 9 [8,-2,4,0,-2,0,0,0,-2] ** 1.000502364766446 81 1973 2904 11 [25,-16,9,-7,2,3,-15,14,-4,-5,-3] ** 1.000198083371338 32 919 2888 12 [29,-8,-3,9,-1,3,-9,1,-3,8,3,-9] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=1/3, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000016313761905 38 1244 17476 6 [34,-43,55,-53,43,-25] ** 1.000042112147912 51 974 16081 9 [20,-18,15,-8,10,-5,-5,10,-12] ** 1.000064117179818 339 1874 11451 7 [7,-9,5,-2,5,-7,3] ** 1.000162039232093 436 1867 9563 8 [5,-3,-1,3,0,-1,-1,1] ** 1.000020717712129 30 1889 9201 4 [63,-3,57,-21] ** 1.000108265319972 103 1167 8318 4 [11,4,-2,-11] ** 1.000152037699776 121 761 8134 6 [6,2,-1,-1,1,-4] ** 1.000268505640947 732 1979 7537 11 [4,-4,1,1,0,-1,1,0,0,-1,1] ** 1.000186457220578 192 1720 6950 5 [9,0,-3,-4,7] ** 1.000036027319351 117 1416 6838 9 [12,1,3,-1,-3,1,3,-1,-3] ** 1.000079918001269 64 374 6478 14 [5,5,0,-3,1,3,-2,-1,2,1,-2,-4,-2,-1] ** 1.000156019281739 191 1211 6314 16 [6,3,-5,-2,-1,-1,-3,0,3,1,1,1,3,0,-3,-1] ** 1.000081935557988 488 1724 6051 10 [3,1,3,0,2,-2,0,-1,-1,-1] ** 1.000329313909862 213 502 6019 17 [3,-2,1,0,1,-1,1,0,1,-1,0,0,0,0,-1,1,-1] ** 1.000329313909862 502 1183 6018 17 [3,-2,1,0,1,-1,1,0,1,-1,0,0,0,0,-1,1,-1] ** 1.000279811372081 229 1413 5571 17 [6,1,0,2,-1,0,-2,0,0,1,0,0,-1,-1,0,0,1] ** 1.000371009548000 575 1430 5287 18 [2,1,1,-1,0,-1,0,-1,0,0,1,0,0,0,1,0,-1,-1] ** 1.000037597110473 30 989 5080 4 [33,-2,30,-11] ** 1.000200789673608 46 538 5003 8 [11,8,2,-5,-7,1,6,1] ** 1.000107364658279 121 943 4930 8 [7,6,1,3,2,-2,-4,-4] ** 1.000242412381303 537 1766 4728 9 [3,0,3,0,2,-2,0,-2,-1] ** 1.000214865252435 88 1816 4560 4 [21,-8,11,-19] ** 1.000185957868797 392 1785 4550 15 [5,-2,0,-1,1,2,0,-1,0,0,-1,0,1,0,-1] ** 1.000454297968062 494 1754 4497 14 [4,-2,2,-2,0,0,0,0,1,0,0,0,0,-1] ** 1.000137217936991 187 1366 4360 5 [7,2,1,4,5] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=2/5, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000032097044243 151 1862 34920 14 [12,4,1,0,-3,0,-3,0,-3,0,-3,-1,8,4] ** 1.000012412932087 41 316 32793 6 [8,-2,-2,-1,3,3] ** 1.000021753465840 76 819 27738 8 [10,8,4,-1,7,8,7,1] ** 1.000084064416757 788 1849 21122 18 [1,2,2,1,1,1,1,0,0,-1,0,0,0,-1,-1,-1,-1,-1] ** 1.000042112147912 51 974 20350 9 [20,-18,15,-8,10,-5,-5,10,-12] ** 1.000020717712129 30 1889 18023 4 [63,-3,57,-21] ** 1.000106871989807 145 1186 15541 7 [8,2,-5,5,3,-1,4] ** 1.000086884923348 62 970 13838 8 [15,10,2,-7,-9,2,8,1] ** 1.000039817648478 156 1734 13331 11 [12,-10,2,-2,4,0,-1,3,-8,9,-1] ** 1.000108548417854 151 1742 10818 4 [12,-5,-5,11] ** 1.000108265319972 103 1167 10000 4 [11,4,-2,-11] ** 1.000037597110473 30 989 9927 4 [33,-2,30,-11] ** 1.000102800010194 309 1835 9387 12 [5,5,3,2,2,2,-2,-3,-1,-1,-3,-4] ** 1.000081935557988 488 1724 8223 10 [3,1,3,0,2,-2,0,-1,-1,-1] ** 1.000186457220578 192 1720 7940 5 [9,0,-3,-4,7] ** 1.000096620275851 277 1909 7935 7 [8,-8,3,-4,4,-6,5] ** 1.000107364658279 121 943 6577 8 [7,6,1,3,2,-2,-4,-4] ** 1.000182159589929 155 863 6569 17 [6,-2,-2,-2,3,0,-2,1,1,-1,0,1,-1,0,1,0,-1] ** 1.000144591346480 37 998 6049 9 [28,-29,36,-28,29,-36,28,-28,8] ** 1.000200789673608 46 538 5934 8 [11,8,2,-5,-7,1,6,1] ** 1.000198409224041 349 1376 5857 8 [4,0,-1,0,2,-1,-2,2] ** 1.000137217936991 187 1366 5705 5 [7,2,1,4,5] ** 1.000284853321768 496 1684 5576 11 [3,1,1,1,-1,-2,0,0,1,0,1] ** 1.000185957868797 392 1785 5572 15 [5,-2,0,-1,1,2,0,-1,0,0,-1,0,1,0,-1] ** 1.000242412381303 537 1766 5477 9 [3,0,3,0,2,-2,0,-2,-1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=1/2, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000002113050868 152 1422 547614 7 [10,-6,0,-3,1,0,4] ** 1.000011829173147 100 1713 84147 6 [18,-15,2,-6,18,-6] ** 1.000013436208326 75 834 64069 10 [12,-10,2,4,2,-4,-1,10,-6,4] ** 1.000012943548178 141 1506 60351 9 [10,7,3,-1,-3,-1,7,4,5] ** 1.000054399161024 244 1460 40035 12 [6,0,-1,3,-3,-1,-4,-2,2,-3,1,4] ** 1.000023097532218 59 954 38697 10 [15,17,29,30,32,27,15,10,2,3] ** 1.000032904252395 38 1892 38619 13 [50,-11,26,10,23,30,6,27,26,10,25,-6,38] ** 1.000021753465840 76 819 37929 8 [10,8,4,-1,7,8,7,1] ** 1.000009119634599 72 1620 35338 5 [23,-11,-6,8,10] ** 1.000012766236563 104 1076 35337 9 [11,-7,2,4,1,-2,-1,1,1] ** 1.000065262933536 387 1579 32975 8 [4,0,1,1,1,1,2,1] ** 1.000020717712129 30 1889 28774 4 [63,-3,57,-21] ** 1.000057021897901 114 645 25959 13 [6,-2,1,-4,2,-1,1,-2,-2,2,0,3,-1] ** 1.000020673026896 169 1862 19088 6 [10,10,13,4,1,-6] ** 1.000039817648478 156 1734 18892 11 [12,-10,2,-2,4,0,-1,3,-8,9,-1] ** 1.000094740320442 150 1862 16454 7 [13,-8,10,-14,10,-8,12] ** 1.000034765624485 205 1299 16298 8 [6,2,1,-1,-1,4,2,3] ** 1.000037597110473 30 989 15889 4 [33,-2,30,-11] ** 1.000062871354546 155 1910 15450 9 [12,4,0,-3,-11,-1,4,5,7] ** 1.000067964806049 172 601 14748 16 [3,1,2,1,3,0,1,1,1,0,0,0,0,1,0,1] ** 1.000109013472929 127 1454 14657 7 [12,-7,9,-13,9,-7,11] ** 1.000108548417854 151 1742 13088 4 [12,-5,-5,11] ** 1.000108265319972 103 1167 12086 4 [11,4,-2,-11] ** 1.000058631039483 419 1971 12052 11 [5,-2,3,-1,2,-2,1,2,0,0,1] ** 1.000083374142554 256 1344 11461 8 [6,-5,6,-2,-2,5,-3,4] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=3/5, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000010533055300 112 1220 99210 9 [11,-2,9,0,5,2,8,-2,7] ** 1.000025958814849 126 1671 51989 12 [13,4,-7,1,-3,-6,-4,-3,3,6,3,-2] ** 1.000006955568155 126 1436 46928 9 [12,-7,1,5,1,-3,-1,1,1] ** 1.000012943548178 141 1506 43137 9 [10,7,3,-1,-3,-1,7,4,5] ** 1.000021753465840 76 819 27655 8 [10,8,4,-1,7,8,7,1] ** 1.000055662303369 53 747 26579 16 [15,-13,3,6,6,-10,6,1,9,-11,5,4,4,-8,8,-4] ** 1.000038363444793 108 886 26008 7 [9,-6,-5,6,-4,-4,6] ** 1.000053944430277 116 1336 23973 6 [12,-6,6,-11,2,1] ** 1.000020717712129 30 1889 18054 4 [63,-3,57,-21] ** 1.000086366805472 73 660 11943 12 [9,0,3,3,1,-1,-2,-1,0,1,1,1] ** 1.000117140587403 121 1323 11934 15 [11,-1,3,1,-2,0,2,0,-1,0,1,0,-1,0,1] ** 1.000060984318025 106 345 11659 11 [3,1,-1,1,2,-2,1,2,0,0,2] ** 1.000060984318025 345 1123 11658 11 [3,1,-1,1,2,-2,1,2,0,0,2] ** 1.000067964806049 172 601 11481 16 [3,1,2,1,3,0,1,1,1,0,0,0,0,1,0,1] ** 1.000009119634599 72 1620 10860 5 [23,-11,-6,8,10] ** 1.000053242638087 326 1922 10239 8 [7,-7,3,-1,3,-6,7,-3] ** 1.000108265319972 103 1167 10040 4 [11,4,-2,-11] ** 1.000037597110473 30 989 9958 4 [33,-2,30,-11] ** 1.000034765624485 205 1299 9887 8 [6,2,1,-1,-1,4,2,3] ** 1.000166574535744 304 952 9526 20 [3,1,-2,0,2,-1,-1,-1,1,0,-1,1,1,-1,-1,1,0,-1,0,1] ** 1.000090095647115 241 1661 9303 14 [7,-1,2,-2,3,0,-2,-4,2,-2,2,-2,3,1] ** 1.000133114927012 228 532 9201 16 [2,1,0,-1,-1,1,0,1,0,0,-1,-1,1,0,0,-1] ** 1.000133114927012 532 1241 9200 16 [2,1,0,-1,-1,1,0,1,0,0,-1,-1,1,0,0,-1] ** 1.000186457220578 192 1720 7944 5 [9,0,-3,-4,7] ** 1.000062941732713 517 1769 7371 9 [4,-2,0,0,1,-1,2,-1,1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=2/3, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000003542706183 142 1924 107470 9 [14,-6,-2,8,-4,5,1,-3,-2] ** 1.000047313152607 334 1398 21282 11 [5,-4,3,-2,-1,1,1,0,1,0,-1] ** 1.000102090094147 100 992 13290 15 [10,-1,2,1,-2,0,2,0,-1,0,1,0,-1,0,1] ** 1.000070737583946 41 469 10628 10 [11,5,1,-8,-1,1,11,-1,-4,-5] ** 1.000117140587403 121 1323 10396 15 [11,-1,3,1,-2,0,2,0,-1,0,1,0,-1,0,1] ** 1.000020717712129 30 1889 9243 4 [63,-3,57,-21] ** 1.000241946612386 22 98 9190 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1] ** 1.000241946612386 98 437 9189 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1] ** 1.000241946612386 437 1949 9188 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1] ** 1.000166574535744 304 952 8457 20 [3,1,-2,0,2,-1,-1,-1,1,0,-1,1,1,-1,-1,1,0,-1,0,1] ** 1.000108265319972 103 1167 8358 4 [11,4,-2,-11] ** 1.000090095647115 241 1661 7293 14 [7,-1,2,-2,3,0,-2,-4,2,-2,2,-2,3,1] ** 1.000142443422050 59 1859 6791 15 [32,-16,16,1,2,4,9,2,5,-6,4,9,-14,4,-8] ** 1.000246850262805 454 1568 5927 10 [2,5,1,-3,-1,1,1,0,-2,-2] ** 1.000065916929107 48 1526 5787 11 [32,-6,-20,4,0,-21,14,18,-3,-10,2] ** 1.000132652778331 289 1837 5749 6 [6,2,1,4,3,-2] ** 1.000270955335396 402 1717 5706 13 [4,1,1,-2,3,-1,0,-2,1,-2,-2,0,1] ** 1.000166590432859 115 1758 5702 13 [15,4,6,0,-4,0,4,2,-2,-2,0,2,2] ** 1.000190145788484 110 686 5219 5 [7,-4,-5,1,3] ** 1.000048962251457 307 1508 5126 7 [5,0,-2,-1,2,2,-4] ** 1.000037597110473 30 989 5111 4 [33,-2,30,-11] ** 1.000346620339653 68 1366 5019 8 [19,21,17,9,-2,-6,-15,-13] ** 1.000234436716968 170 1062 4818 8 [6,2,-3,1,1,-4,1,5] ** 1.000081284932752 50 295 4676 9 [6,-1,3,-3,0,-2,-1,2,-2] ** 1.000081284932752 295 1741 4675 9 [6,-1,3,-3,0,-2,-1,2,-2] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=3/4, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000118862375490 267 1605 11953 7 [5,6,1,-3,-1,0,-2] ** 1.000098076956125 235 563 5705 11 [2,1,0,0,-1,1,0,-1,-1,0,1] ** 1.000098076956125 563 1349 5704 11 [2,1,0,0,-1,1,0,-1,-1,0,1] ** 1.000108265319972 103 1167 5664 4 [11,4,-2,-11] ** 1.000270955335396 402 1717 4646 13 [4,1,1,-2,3,-1,0,-2,1,-2,-2,0,1] ** 1.000093030795381 136 440 4085 8 [3,1,-1,1,0,-2,-1,1] ** 1.000093030795381 440 1424 4084 8 [3,1,-1,1,0,-2,-1,1] ** 1.000305043968183 134 1623 3803 16 [13,-11,3,0,-2,-2,-1,1,1,1,0,-1,-1,0,1,1] ** 1.000202028544894 81 542 3728 11 [7,-2,0,-3,2,-1,-1,-4,1,2,1] ** 1.000132652778331 289 1837 3569 6 [6,2,1,4,3,-2] ** 1.000314732063239 36 1085 3554 13 [30,4,6,-4,-5,-1,5,1,-5,-1,5,1,-5] ** 1.000616540428395 320 1279 3202 17 [4,0,0,0,-1,1,-1,1,-1,1,-1,1,0,-3,0,0,1] ** 1.000962235223722 315 768 2857 10 [3,-2,2,-1,0,-1,1,-1,1,-1] ** 1.000962235223722 768 1873 2856 10 [3,-2,2,-1,0,-1,1,-1,1,-1] ** 1.000815453738719 97 1052 2458 17 [10,9,2,-1,0,0,1,4,-2,-1,-3,-7,-4,0,-2,-4,2] ** 1.000659710741214 96 684 2435 8 [7,1,0,-5,-2,0,4,-2] ** 1.000539486988851 310 814 2409 9 [2,2,-1,0,1,-1,-1,0,1] ** 1.000642177222389 182 609 2375 9 [4,-3,3,-1,0,1,-1,1,-2] ** 1.000121054039365 22 93 2324 11 [4,1,0,-1,1,0,0,-3,0,-1,1] ** 1.000121054039365 93 393 2323 11 [4,1,0,-1,1,0,0,-3,0,-1,1] ** 1.000121054039365 393 1661 2322 11 [4,1,0,-1,1,0,0,-3,0,-1,1] ** 1.000550715091592 112 1830 2229 6 [16,5,9,-2,7,-7] ** 1.000464938518246 133 1057 2219 10 [9,-9,6,-8,6,-7,5,-2,6,-4] ** 1.000379615738119 64 1457 2174 4 [23,-6,16,-16] ** 1.000701863506824 247 966 2049 21 [5,-5,3,0,-2,2,-1,0,1,-1,1,-1,0,1,-1,1,0,-1,1,-1,1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=4/5, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000230561690733 587 1655 9512 11 [2,2,1,0,0,-2,-2,-1,0,1,1] ** 1.000216742303556 93 1010 3622 15 [11,-2,5,2,9,-3,5,-2,-2,-2,-3,-5,-4,1,-7] ** 1.000588628388860 205 439 3344 20 [3,-2,1,-2,1,1,-1,1,-2,2,-2,2,-1,1,-1,-1,2,-1,1,-1] ** 1.000588628388860 439 940 3343 20 [3,-2,1,-2,1,1,-1,1,-2,2,-2,2,-1,1,-1,-1,2,-1,1,-1] ** 1.000498435859590 580 1483 2852 15 [3,-1,0,-1,0,1,0,-1,0,1,0,0,-1,-1,1] ** 1.000755504545288 372 1126 2703 15 [3,0,1,-2,-1,0,2,0,0,-2,-1,1,0,1,-1] ** 1.000202028544894 81 542 2595 11 [7,-2,0,-3,2,-1,-1,-4,1,2,1] ** 1.000465887867211 416 1393 2399 13 [3,1,1,-1,-1,-2,1,-1,1,-1,0,0,1] ** 1.000369622285660 652 1547 2335 17 [3,-1,-1,-1,1,2,-2,0,0,0,1,-1,0,0,0,1,-1] ** 1.000727451123105 150 1721 2267 15 [11,5,5,0,-3,-4,-12,0,-1,1,-1,0,1,-1,1] ** 1.000659710741214 96 684 2077 8 [7,1,0,-5,-2,0,4,-2] ** 1.000607401714680 326 856 2040 10 [3,-1,0,0,1,-2,1,-1,1,-1] ** 1.000177346113274 44 1648 2024 10 [39,-59,42,-10,-9,19,-10,-9,20,-17] ** 1.000390540272179 610 1596 1999 15 [3,-1,0,0,0,0,0,-1,0,0,0,0,0,1,-1] ** 1.000642177222389 182 609 1992 9 [4,-3,3,-1,0,1,-1,1,-2] ** 1.000832153882265 297 1732 1910 19 [6,-1,0,1,-2,1,0,1,-2,1,1,-1,0,-4,0,0,-1,2,-1] ** 1.000132652778331 289 1837 1891 6 [6,2,1,4,3,-2] ** 1.000550715091592 112 1830 1812 6 [16,5,9,-2,7,-7] ** 1.000659981878357 266 1084 1766 6 [5,-4,1,-1,4,-3] ** 1.000968314086598 200 614 1659 14 [3,0,1,-1,0,-1,2,0,0,-1,-1,-1,1,-1] ** 1.000968314086598 614 1885 1658 14 [3,0,1,-1,0,-1,2,0,0,-1,-1,-1,1,-1] ** 1.000379615738119 64 1457 1628 4 [23,-6,16,-16] ** 1.000734414794903 482 1418 1493 8 [3,-1,3,-2,2,-3,1,-1] ** 1.000134179125879 91 914 1484 13 [11,-10,4,0,-2,-2,-1,1,1,1,0,0,-1] ** 1.000699186097681 105 396 1460 15 [3,3,0,-1,-1,1,1,-1,-1,1,1,0,-1,-3,-1] ** ----------------- ---- ---- ------ -- ------------ ** ** 25 longest fake sequences for r=1, 2 <= x <= 2000, x+2 <= y <= 2000 ** root x y fail o coefficients ** ----------------- ---- ---- ------ -- ------------ ** 1.000028402156455 450 1695 38415 14 [4,-1,1,-2,0,-1,3,-2,0,0,0,0,1,-2] ** 1.000190482197144 124 390 13414 13 [4,-3,1,0,0,0,-1,1,0,-1,0,2,-2] ** 1.000190482197144 390 1227 13413 13 [4,-3,1,0,0,0,-1,1,0,-1,0,2,-2] ** 1.000143584240061 275 1330 6504 10 [5,-1,1,0,1,-2,-1,-1,-2,1] ** 1.000437651448089 597 1940 4341 10 [4,-3,2,0,-2,1,-1,-1,2,-1] ** 1.000544776855336 25 84 4099 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2] ** 1.000544776855336 84 283 4098 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2] ** 1.000544776855336 283 954 4097 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2] ** 1.000304050189770 324 1304 3739 9 [3,4,1,-1,-3,-3,-2,0,2] ** 1.000257081176362 92 548 3379 8 [5,5,4,2,-3,-6,-4,-2] ** 1.000438783587243 605 1577 2856 12 [3,-1,0,0,-1,2,-1,-1,0,0,1,-1] ** 1.000655507570736 176 587 2401 8 [3,1,1,-1,-3,-1,0,1] ** 1.000655507570736 587 1958 2400 8 [3,1,1,-1,-3,-1,0,1] ** 1.000890787851308 374 1506 2387 9 [4,0,1,-2,-1,0,1,-1,-1] ** 1.000689380300664 81 269 2261 12 [4,-2,-1,1,-2,3,-2,0,1,-1,2,-2] ** 1.000689380300664 269 894 2260 12 [4,-2,-1,1,-2,3,-2,0,1,-1,2,-2] ** 1.000196961275789 642 1965 2232 12 [3,1,-2,-2,1,2,0,-3,0,1,1,-1] ** 1.000930815480001 420 1766 2004 14 [5,-4,3,-1,0,0,1,-1,-1,2,-4,3,-3,1] ** 1.000916107354094 319 1505 1815 10 [5,-1,-2,2,1,-2,-1,2,0,-3] ** 1.000605568483049 170 1790 1802 11 [10,5,6,1,-2,0,3,-3,-4,-8,-7] ** 1.000817823206005 206 495 1756 10 [3,-1,-1,0,-1,2,0,-1,1,-1] ** 1.000817823206005 495 1190 1755 10 [3,-1,-1,0,-1,2,0,-1,1,-1] ** 1.001326318073690 458 1409 1617 10 [3,1,-2,-1,0,-1,0,3,0,-2] ** 1.000499490255901 165 425 1613 12 [3,-1,0,-1,1,0,1,-2,1,-1,1,-1] ** 1.000499490255901 425 1095 1612 12 [3,-1,0,-1,1,0,1,-2,1,-1,1,-1] ** ----------------- ---- ---- ------ -- ------------ ** */