有趣的数字规律

Elivs

已于 2025-02-25 08:21:49 修改

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文章标签：算法

于 2025-02-20 15:41:59 首次发布

本文链接：https://blog.youkuaiyun.com/Elivs/article/details/145755529

版权

-- 简单的算数（一维直线）

-- 勾股定理（二维平面）

-- 三维空间

这个误差，也只比著名的哈代-拉马努金数的误差小1。哈代-拉马努金数是。误差是1

费马大定理指出，没有三个正整数 a、b 和 c 满足方程，其中 n 是大于2的整数。

但就和来说，似乎a=2n-1, b=2n, c=2n+1时，非常接近于。

所以我猜想可以当做费马大定理的一个近似解。

用计算机算了n从1到100，与之间的差不超过0.25%。

当然，n越大，误差率会随着增长。

用二项式定理以及欧拉常数e的定义，可以证明n趋于无穷时，不等式左边约相当于右边的百分之97点5。

N	a	b	c					ERROR	ratio
1	1	2	3	1	2	3	3	0	0
2	3	4	5	9	16	25	25	0	0
3	5	6	7	125	216	343	341	2	0.005831
4	7	8	9	2401	4096	6561	6497	64	0.009755
5	9	10	11	59049	100000	161051	159049	2002	0.012431
6	11	12	13	1771561	2985984	4826809	4757545	69264	0.01435
7	13	14	15	62748517	105413504	170859375	168162021	2697354	0.015787
8	15	16	17	2562890625	4294967296	6975757441	6857857921	117899520	0.016901
9	17	18	19	1.18588E+11	1.98359E+11	3.22688E+11	3.16947E+11	5740530914	0.01779
10	19	20	21	6.13107E+12	1.024E+13	1.66799E+13	1.63711E+13	3.08815E+11	0.018514
11	21	22	23	3.50278E+14	5.84318E+14	9.5281E+14	9.34596E+14	1.8214E+13	0.019116
12	23	24	25	2.19146E+16	3.65203E+16	5.96046E+16	5.8435E+16	1.16967E+15	0.019624
13	25	26	27	1.49012E+18	2.48115E+18	4.05256E+18	3.97127E+18	8.12862E+16	0.020058
14	27	28	29	1.09419E+20	1.82059E+20	2.97558E+20	2.91478E+20	6.08012E+18	0.020433
15	29	30	31	8.62919E+21	1.43489E+22	2.34653E+22	2.29781E+22	4.87166E+20	0.020761
16	31	32	33	7.27423E+23	1.20893E+24	1.97799E+24	1.93635E+24	4.16363E+22	0.02105
17	33	34	35	6.52735E+25	1.08428E+26	1.77483E+26	1.73702E+26	3.78145E+24	0.021306
18	35	36	37	6.2119E+27	1.03144E+28	1.68901E+28	1.65263E+28	3.63724E+26	0.021535
19	37	38	39	6.24932E+29	1.03726E+30	1.69913E+30	1.66219E+30	3.69399E+28	0.02174
20	39	40	41	6.62662E+31	1.09951E+32	1.80168E+32	1.76217E+32	3.95041E+30	0.021926
21	41	42	43	7.38688E+33	1.22528E+34	2.00834E+34	1.96397E+34	4.43743E+32	0.022095
22	43	44	45	8.63587E+35	1.43205E+36	2.34788E+36	2.29564E+36	5.22376E+34	0.022249
23	45	46	47	1.05654E+38	1.75158E+38	2.87244E+38	2.80813E+38	6.43133E+36	0.02239
24	47	48	49	1.35005E+40	2.23764E+40	3.67034E+40	3.58768E+40	8.26534E+38	0.022519
25	49	50	51	1.79847E+42	2.98023E+42	4.88939E+42	4.7787E+42	1.1069E+41	0.022639
26	51	52	53	2.49359E+44	4.1313E+44	6.77911E+44	6.62489E+44	1.5422E+43	0.022749
27	53	54	55	3.59293E+46	5.95157E+46	9.76771E+46	9.5445E+46	2.2321E+45	0.022852
28	55	56	57	5.37224E+48	8.89743E+48	1.46048E+49	1.42697E+49	3.35139E+47	0.022947
29	57	58	59	8.32474E+50	1.37852E+51	2.26312E+51	2.21099E+51	5.21337E+49	0.023036
30	59	60	61	1.33524E+53	2.21074E+53	3.6299E+53	3.54598E+53	8.39209E+51	0.023119
31	61	62	63	2.21424E+55	3.66558E+55	6.01945E+55	5.87982E+55	1.39635E+54	0.023197
32	63	64	65	3.79226E+57	6.2771E+57	1.03093E+58	1.00694E+58	2.39901E+56	0.02327
33	65	66	67	6.70102E+59	1.10905E+60	1.82166E+60	1.77915E+60	4.25163E+58	0.023339
34	67	68	69	1.22052E+62	2.01978E+62	3.31794E+62	3.24029E+62	7.76536E+60	0.023404
35	69	70	71	2.28938E+64	3.78819E+64	6.22361E+64	6.07757E+64	1.46039E+63	0.023465
36	71	72	73	4.41876E+66	7.31088E+66	1.20122E+67	1.17296E+67	2.82566E+65	0.023523
37	73	74	75	8.76891E+68	1.45069E+69	2.38378E+69	2.32758E+69	5.6205E+67	0.023578
38	75	76	77	1.78784E+71	2.95744E+71	4.86013E+71	4.74528E+71	1.14845E+70	0.02363
39	77	78	79	3.7423E+73	6.18998E+73	1.01732E+74	9.93228E+73	2.40895E+72	0.023679
40	79	80	81	8.03681E+75	1.32923E+76	2.18475E+76	2.13291E+76	5.18361E+74	0.023726
41	81	82	83	1.76964E+78	2.92663E+78	4.81063E+78	4.69627E+78	1.14354E+77	0.023771
42	83	84	85	3.99282E+80	6.60283E+80	1.08541E+81	1.05956E+81	2.58477E+79	0.023814
43	85	86	87	9.22601E+82	1.52557E+83	2.508E+83	2.44818E+83	5.98267E+81	0.023854
44	87	88	89	2.18196E+85	3.60776E+85	5.93144E+85	5.78972E+85	1.41721E+84	0.023893
45	89	90	91	5.27898E+87	8.72796E+87	1.43504E+88	1.40069E+88	3.43408E+86	0.02393
46	91	92	93	1.30588E+90	2.15894E+90	3.5499E+90	3.46482E+90	8.50763E+88	0.023966
47	93	94	95	3.3014E+92	5.45769E+92	8.97448E+92	8.7591E+92	2.15387E+91	0.024
48	95	96	97	8.52576E+94	1.40935E+95	2.31763E+95	2.26193E+95	5.56985E+93	0.024033
49	97	98	99	2.2481E+97	3.71602E+97	6.11117E+97	5.96411E+97	1.47059E+96	0.024064
50	99	100	101	6.0501E+99	1E+100	1.6446E+100	1.605E+100	3.96258E+98	0.024094
51	101	102	103	1.6611E+102	2.7454E+102	4.5154E+102	4.4065E+102	1.0893E+101	0.024123
52	103	104	105	4.6509E+104	7.6866E+104	1.2643E+105	1.2337E+105	3.0533E+103	0.024151
53	105	106	107	1.3275E+107	2.1939E+107	3.6086E+107	3.5214E+107	8.7248E+105	0.024178
54	107	108	109	3.8612E+109	6.3809E+109	1.0496E+110	1.0242E+110	2.5404E+108	0.024203
55	109	110	111	1.1441E+112	1.8906E+112	3.11E+112	3.0347E+112	7.5351E+110	0.024228
56	111	112	113	3.4521E+114	5.7044E+114	9.3841E+114	9.1565E+114	2.2759E+113	0.024252
57	113	114	115	1.0604E+117	1.7522E+117	2.8826E+117	2.8126E+117	6.9975E+115	0.024276
58	115	116	117	3.3149E+119	5.4773E+119	9.0111E+119	8.7922E+119	2.1895E+118	0.024298
59	117	118	119	1.0543E+122	1.742E+122	2.866E+122	2.7963E+122	6.9699E+120	0.02432
60	119	120	121	3.4105E+124	5.6348E+124	9.2709E+124	9.0452E+124	2.2566E+123	0.02434
61	121	122	123	1.1218E+127	1.8533E+127	3.0494E+127	2.9751E+127	7.4285E+125	0.024361
62	123	124	125	3.7507E+129	6.1965E+129	1.0196E+130	9.9472E+129	2.4858E+128	0.02438
63	125	126	127	1.2745E+132	2.1054E+132	3.4645E+132	3.3799E+132	8.453E+130	0.024399
64	127	128	129	4.3999E+134	7.2684E+134	1.196E+135	1.1668E+135	2.9204E+133	0.024418
65	129	130	131	1.5429E+137	2.5487E+137	4.1941E+137	4.0916E+137	1.0248E+136	0.024436
66	131	132	133	5.4942E+139	9.0757E+139	1.4935E+140	1.457E+140	3.6521E+138	0.024453
67	133	134	135	1.9864E+142	3.2811E+142	5.3996E+142	5.2675E+142	1.3213E+141	0.02447
68	135	136	137	7.2895E+144	1.2041E+145	1.9815E+145	1.933E+145	4.852E+143	0.024486
69	137	138	139	2.7147E+147	4.4839E+147	7.3794E+147	7.1986E+147	1.8081E+146	0.024502
70	139	140	141	1.0257E+150	1.6942E+150	2.7883E+150	2.7199E+150	6.8361E+148	0.024517
71	141	142	143	3.9315E+152	6.4934E+152	1.0687E+153	1.0425E+153	2.6218E+151	0.024532
72	143	144	145	1.5283E+155	2.5241E+155	4.1543E+155	4.0523E+155	1.0197E+154	0.024547
73	145	146	147	6.0237E+157	9.9485E+157	1.6374E+158	1.5972E+158	4.0217E+156	0.024561
74	147	148	149	2.407E+160	3.9753E+160	6.5431E+160	6.3823E+160	1.6079E+159	0.024575
75	149	150	151	9.7492E+162	1.6101E+163	2.6502E+163	2.585E+163	6.5162E+161	0.024588
76	151	152	153	4.0017E+165	6.6086E+165	1.0878E+166	1.061E+166	2.6761E+164	0.024601
77	153	154	155	1.6643E+168	2.7485E+168	4.5242E+168	4.4128E+168	1.1136E+167	0.024614
78	155	156	157	7.0125E+170	1.158E+171	1.9062E+171	1.8593E+171	4.6943E+169	0.024626
79	157	158	159	2.9928E+173	4.9421E+173	8.1353E+173	7.9348E+173	2.0044E+172	0.024638
80	159	160	161	1.2935E+176	2.136E+176	3.5162E+176	3.4295E+176	8.6674E+174	0.02465
81	161	162	163	5.661E+178	9.3479E+178	1.5388E+179	1.5009E+179	3.7951E+177	0.024662
82	163	164	165	2.5083E+181	4.1419E+181	6.8184E+181	6.6502E+181	1.6823E+180	0.024673
83	165	166	167	1.125E+184	1.8577E+184	3.0582E+184	2.9827E+184	7.5488E+182	0.024684
84	167	168	169	5.1072E+186	8.433E+186	1.3883E+187	1.354E+187	3.4283E+185	0.024695
85	169	170	171	2.3462E+189	3.874E+189	6.3778E+189	6.2202E+189	1.5756E+188	0.024705
86	171	172	173	1.0906E+192	1.8007E+192	2.9646E+192	2.8913E+192	7.3271E+190	0.024715
87	173	174	175	5.1287E+194	8.4681E+194	1.3942E+195	1.3597E+195	3.4471E+193	0.024725
88	175	176	177	2.4398E+197	4.0282E+197	6.632E+197	6.468E+197	1.6404E+196	0.024735
89	177	178	179	1.1739E+200	1.9381E+200	3.1909E+200	3.112E+200	7.8958E+198	0.024744
90	179	180	181	5.7118E+202	9.4303E+202	1.5526E+203	1.5142E+203	3.8434E+201	0.024754
91	181	182	183	2.8103E+205	4.6398E+205	7.6392E+205	7.4501E+205	1.8917E+204	0.024763
92	183	184	185	1.398E+208	2.308E+208	3.8001E+208	3.706E+208	9.4136E+206	0.024772
93	185	186	187	7.0302E+210	1.1607E+211	1.911E+211	1.8637E+211	4.7357E+209	0.024781
94	187	188	189	3.5736E+213	5.8998E+213	9.7143E+213	9.4734E+213	2.4081E+212	0.024789
95	189	190	191	1.836E+216	3.031E+216	4.9908E+216	4.867E+216	1.2376E+215	0.024797
96	191	192	193	9.5324E+218	1.5737E+219	2.5912E+219	2.5269E+219	6.4277E+217	0.024806
97	193	194	195	5.001E+221	8.256E+221	1.3594E+222	1.3257E+222	3.3733E+220	0.024814
98	195	196	197	2.6509E+224	4.3762E+224	7.2059E+224	7.0271E+224	1.7886E+223	0.024822
99	197	198	199	1.4196E+227	2.3434E+227	3.8588E+227	3.763E+227	9.5812E+225	0.024829
100	199	200	201	7.6791E+229	1.2677E+230	2.0874E+230	2.0356E+230	5.1844E+228	0.024837