Fermat near-misses

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 |x³−y²| 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 C_nlog(N)/r₀, where C_n is the area of the l_n “circle” |x|ⁿ+|y|ⁿ<1; and that r≪z^o(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 CN^1/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 = 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The table for Hall’s conjecture (solutions of 0 < |x³−y²| < x^1/2 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

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

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

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

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