## Tables of Fermat “near-misses” approximate solutions of xn + yn = zn in integers with 0 < x <= y < z < 223 and n in [4,20]

Tables from my paper “Rational points near curves and small nonzero |x3y2| via lattice reduction”, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 (math.NT/0005139 on the arXiv), extended from z<106 to z<223=8388608, and with the threshold on the absolute value of

r = n zn−3 / (zn − yn − xn)
lowered from 4 to 1. This computation took most of a week on a 500MHz machine running GP.

The usual heuristics suggest that for any n and r0 the number of solutions of |r|<r0 with z<N should be asymptotically proportional to Cnlog(N)/r0, where Cn is the area of the ln “circle” |x|n+|y|n<1; and that r≪zo(1) should hold for each n>3. More precisely, these heuristics apply to “random” solutions; there can also be systematic families of solutions that exceed random expectations. A simple example of such a family is

(2ntn+1)n − (2ntn−1)n = (2ntn−1)n + (2n/24)(n−2)(n−1)nn−2t(n−3)n + O(t(n−5)n),
with r = 3/((n−2)(n−1)) + O(t−n). Thus we have at least CN1/n cases of |r|>3/n2 with z<N, more than the expected logarithmic growth. We do not see this family in our computations, because 3/((n−2)(n−1)) is below our threshold of 1. But some of the more striking Fermat near-misses that were revealed by the computation generalize to less-obvious families in which r grows as a positive power of z. One such family appears for n=8:
(32 t9 + 6 t)8 + (32 t8 + 7)8 = (32 t9 + 10 t)8 + 21·228 t40 + O(t32),
with r = t5/21 + O(t−3). More generally, there is such a family whenever 3n(n−2) is a square, that is n=(2, 3,) 8, 27, 98, 363,... (see Theorem 5 in my ANTS-4 paper). The tables also include the first few representatives of the following family for n=4, which I didn’t notice until extending the computation from 106 to 223:
(192 t8 + 24 t4 − 1)4 − (192 t8 − 24 t4 − 1)4 − (192 t7)4 = −192 t4.
This yields r = −4t4 + O(1) = (z/12)1/2 + O(1). It also gives a rational curve on the twisted Fermat quartic surface x4+y4=z4+12w4 that seems to be previously unknown. Once we see this trick, we readily generalize to rational curves on twisted Fermat hypersurfaces of degree n>4 in projective (n−2)-space (that is, linear combinations of n nth powers set to zero); for instance, for n=5, start from the identity
(u2 + 2 u − 2)5 − (u2 − 2 u − 2)5 = 20 u9 − 96 u5 + 320 u,
and make u some multiple of t5.

Besides polynomial families, there is another systematic effect that distinguishes the distribution of 1/r values from randomness: if (x,y,z) yields a ratio of r for some n, then (kx,ky,kz) yields r/k^3 for each k=2,3,4,... When |r|>8 and z<222, at least one such “aftershock” ratio appears in the tables below.

The occurrences of |r|>10, other than those in known polynomial families, are as follows:
 r = − 184.1: 34720737 + 46270117 = 1.00000000000000000000036... · 47108687 r = 137.1: 28010 + 30510 = 0.999999997... · 31610 r = − 120.4: 135 + 165 = 175 + 12 (as it happens also 13+16=17+12...) r = 78.4: 3866927 + 4114137 = 0.9999999999999999989... · 4418497 r = 42.9: 43443715 + 58812915 = 0.999999999999999998... · 58854415 r = 36.9: 8444875 + 12884395 = 13182025 − 235305158626 = 0.99999999999999999994... · 13182025 r = 24.7: 218609 + 252089 = 0.999999999999979... · 259039 r = 18.0: 3286586 + 15072536 = 0.99999999999999999990... · 15072806 r = 14.8: 762154 + 3113904 = 3116694 − 84096 = 0.999999999999999991... · 3116694 r = −11.7: 180366416 + 229856516 = 1.00000000000000000011... · 230150516 r = 11.0: 85833910 + 216235910 = 0.99999999999999999991... · 216238010 r = 10.3: 78176912 + 85272312 = 0.999999999999999998... · 87445612
The first of these also has the smallest unnormalized relative error |zn − yn − xn| / zn in the tables, namely 3.64 · 10−22.

### Tables for n = 4567891011121314151617181920

The table for Hall’s conjecture (solutions of 0 < |x3y2| < x1/2 in integers with x<1018) is here.

### n=4

 x y z r 21 36 37 2.3 167 192 215 −4.5 242 471 479 1.7 717 967 1033 −1.1 2111 2285 2620 2.4 8191 16253 16509 12.9 16893 27753 28660 1.4 16382 32506 33018 1.6 24576 48767 49535 −64.5 64493 85903 92037 2.4 49152 97534 99070 −8.1 74191 118430 122748 3.7 73728 146301 148605 −2.4 34231 157972 158059 5.2 118138 189321 196122 1.4 98304 195068 198140 −1.0 76215 311390 311669 14.8 228730 398176 408599 2.0 249308 288317 322171 −1.4 152430 622780 623338 1.9 419904 1257767 1261655 −324.5 468750 1168121 1175621 −2.4 1259712 3773301 3784965 −12.0 839808 2515534 2523310 −40.6 2519424 7546602 7569930 −1.5 2099520 6288835 6308275 −2.6 1679616 5031068 5046620 −5.1 2652287 5948932 6006844 1.2

### n=5

 x y z r 13 16 17 −120.4 26 32 34 −15.1 39 48 51 −4.5 52 64 68 −1.9 42 71 72 −8.8 104 133 140 1.3 84 142 144 −1.1 133 228 231 3.0 262 328 347 −6.2 494 954 961 1.9 1125 2335 2347 −5.0 5088 16155 16165 4.1 4572 13234 13247 −2.7 8164 13595 13801 −1.2 9225 12596 13087 −1.3 14447 54751 54765 −2.3 190512 292329 298900 5.5 464726 871287 878683 −1.0 844487 1288439 1318202 36.9 2533461 3865317 3954606 1.4 1688974 2576878 2636404 4.6 2010324 2220733 2442021 1.7 707902 5645541 5645576 −4.0

### n=6

 x y z r 428 643 652 −1.6 6107 8919 9066 −9.9 12214 17838 18132 −1.2 185256 325912 327719 −1.2 197687 324859 327552 −1.1 584496 613841 673549 1.1 546180 561811 622148 4.3 311426 1760135 1760144 1.9 328658 1507253 1507280 18.0 943766 1625656 1635867 −1.4 657316 3014506 3014560 2.3 2463585 4820496 4834706 1.0

### n=7

 x y z r 173 216 222 −1.1 1497 2797 2802 2.4 3029 3964 4045 1.5 13226 38721 38724 1.9 16607 36503 36524 3.5 22754 69507 69511 1.1 39235 71117 71274 1.2 56817 85813 86481 −1.3 181914 187234 203913 −1.1 386692 411413 441849 78.4 479723 728924 734369 3.8 773384 822826 883698 9.8 1546768 1645652 1767396 1.2 1160076 1234239 1325547 2.9 1636259 7461304 7461330 1.3 3472073 4627011 4710868 −184.1 6212317 6790851 7220355 1.1 4187720 4206429 4634015 1.5

### n=8

 x y z r 11 11 12 1.6 453 453 494 −1.4 673 845 861 2.3 526 956 957 −1.5 868 1144 1159 −3.5 3673 5266 5302 1.1 8185 16364 16372 1.5 8199 16396 16404 −1.5 29698 62189 62210 2.4 209959 629874 629886 −11.6 209945 629826 629838 11.6 419918 1259748 1259772 −1.4 419890 1259652 1259676 1.4 1744473 7504204 7504212 −1.5 2097145 8388568 8388584 48.8 1647102 5764849 5764881 −2.1 1647070 5764721 5764753 2.1

### n=9

 x y z r 68 69 74 1.2 279 392 394 1.1 6817 10727 10747 5.3 21860 25208 25903 24.7 43720 50416 51806 3.1 51454 105711 105729 2.1 53490 69811 70490 2.5 3036526 3796741 3850115 −1.5

### n=10

 x y z r 36 41 42 1.7 44 47 49 −1.6 280 305 316 137.1 419 462 477 −5.0 560 610 632 17.1 840 915 948 5.1 1400 1525 1580 1.1 1120 1220 1264 2.1 2140 2704 2729 −2.1 7533 8834 8999 4.4 25823 30716 31219 1.0 52693 55961 58460 1.6 858339 2162359 2162380 11.0 1297199 2338623 2339267 −1.5 1943805 2687763 2698102 −2.5 1744769 2143231 2169172 1.3 1716678 4324718 4324760 1.4 4677066 7625650 7631375 1.5

### n=11

 x y z r 123 123 131 −1.1 279 385 386 2.4 301 390 392 1.0 1264 1280 1355 1.1 2722 3509 3528 −2.3 32567 36078 37011 1.8 54065 70427 70768 −2.0 109105 195532 195561 1.4 277501 508559 508618 −1.1 855136 2094587 2094597 1.1 1046373 2260294 2260337 −1.6 2701163 3867055 3873786 −2.0 2295724 5641233 5641259 3.4 3073792 4798307 4801548 −1.1

### n=12

 x y z r 1363 1808 1813 −1.5 1782 1841 1922 6.1 3987 4365 4472 −7.1 22723 32478 32515 −3.9 72837 102740 102877 −3.0 106751 215106 215110 −2.9 130765 241124 241137 −3.5 781769 852723 874456 10.3 680546 1113804 1114055 2.0 1395408 1863675 1868429 −1.4 1563538 1705446 1748912 1.3

### n=13

 x y z r 84 88 91 −2.0 204 209 218 2.0 666 806 811 8.3 1332 1612 1622 1.0 2228 2943 2949 −2.2 5579 8235 8239 4.1 312816 338207 346352 1.1 690178 891066 893501 −2.1 1193321 2755200 2755204 −4.7 1434569 2492169 2492315 2.9

### n=14

 x y z r 4433 4519 4706 −1.1 743276 878164 883981 −3.6 1636734 2037442 2044083 1.6 1441026 2675720 2675753 −4.0 5418289 6182083 6247134 3.4

### n=15

 x y z r 73253 113214 113225 3.9 260149 308540 310077 −2.9 434437 588129 588544 42.9 620289 1153397 1153404 4.5 868874 1176258 1177088 5.4 1303311 1764387 1765632 1.6 2313826 2485665 2534822 −1.3 5127050 6576051 6586417 −2.3 5692275 6930706 6954254 9.7

### n=16

 x y z r 1457 1628 1644 1.0 4546 5231 5264 −1.3 24180 30683 30725 1.0 173859 228771 228947 −1.8 492151 741267 741333 4.6 1503801 2409065 2409145 2.5 1803664 2298565 2301505 −11.8 2296553 2580018 2603406 −3.6 3607328 4597130 4603010 −1.5

### n=17

 x y z r 24 24 25 1.3 93 95 98 −1.1 7315 9459 9466 2.0 15164 20964 20969 2.4 1998569 2248474 2265278 2.6 5567975 7624098 7626238 4.6

### n=18

 x y z r 3792 4868 4871 1.3 2662440 4129507 4129592 3.3

### n=19

 x y z r 79 85 86 −4.7 491 565 567 4.9 2582 3041 3048 2.1 43329 51144 51257 5.8 3441746 5479099 5479141 2.1

### n=20

 x y z r 4110 4693 4709 4.3 17764 22616 22625 −1.3 35816 37074 37835 −1.7 255738 347841 347878 2.5 852068 866702 890301 1.3 2674998 3567225 3567788 −1.4 2593096 2880825 2897442 6.1 3198945 4913429 4913475 −1.2 7706288 7937911 8114575 1.1