Exponential diophantine problems Gottfried Helms - Univ Kassel          (Miniatures since 07' 2007)

 

Table for Fermat-quotients of degree m≥2
for integer bases b<p (where p ∈ ℙ )

The following table displays p-adic-expansions of bases b, for which

               bp-1 ≡ 1 (mod pm)

In ([1],2009) I discussed earlier the problem of Fermat-quotients and introduced a couple of small tools and concepts which I use here. This table focuses on data for only a specific subset of the entire problem.

The casual reader who is not aware of the concept of Fermat-quotient may look in the wikipedia (see:[wp]) first which gives some basic insight and a couple of further references/online-links.

The reader may find some quite involved related tables in the internet, see the references-section at the end of this article for some online-links to homepages which I got aware of.

 

1) In the following I (mis)use the braces for a shorter notation to express the above statement for a more general view. (A misuse which has been helpful through all my discussions/treatizes of exponential diophantine problems.) For the case, that a number n contains the primefactor p to the m'th power I write for p's canonical exponent in n:

{n, p} = m                                          ==> n = x ∙ pm ∙with gcd(x,p)=1

                Note that this notation is a bit more algebraic/concise than "pm| n" or "n≡0 (mod pm)"because -if in the two latter two cases m is the indeterminate- then the expression gives only an upper limit and not a precise value to m. So for instance,

{2p-1-1, p} = 2

is a bit more restricted/precise than the original definition for a Wieferich-prime p which requires only

{2p-1-1, p} ≥ 2

(where we know only two solutions p=1093 and p=3511 for m ≥2 anyway).

Similary for instance

{35-1 , 11} = 2
{196-1 , 7} = 3
{7408626 – 1,7}  = 7
{17590908883812 – 1,13} = 12

indicate Fermat-quotients for (3, 11), (19, 7), (740862, 7) and (175909088838, 13) of degrees 2,3,7 and 12 respectively.

 

 

2) Let's define here the term "degree of the Fermat-quotient":

{bp-1-1, p } = m                  ==> m is the degree of the Fermat-quotient of the pair (b,p)

 

 

3) By the p-adic-representation of the bases b which provide high Fermat-quotients for some prime p we can read the table like in the following example:

p	b	d0	d1	d2	d3	d4	d4	d5	d6	d7	d8	d9	d10	...
7	1	1	0	0	0	0	0	0	0	0	0	0	0	...
7	2	2	4	6	3	0	2	6	2	4	3	4	4	...
7	3	3	4	6	3	0	2	6	2	4	3	4	4	...
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...

 

Here the column p indicates the prime, the column b the "root" bases - "root" means here, that from b = d₀ further bases b_m can be derived according to the given p-adic digits d₀,d₁,d₂,... Actually the entry in that column is identical with the entry in column d₀ which simply contain all congruence-classes (mod p) .

The "p-adic" convention means, that the "digits" are just the digits of the number-representation of some number in the number-system to base p, so to say:

b_m = "d_m-1 ... d₂ d₁ d₀" _{base p}

so for the row with "root" base b=2 we can get the following subsequent bases b₁,b₂,...

b₁ = 2                                   = 2                         ==>        {26 -1, 7} = 1
b₂ = 2 + 4∙7                        = 30                      ==>        {306-1, 7} = 2
b₃ = 2 + 4∙7+ 6∙72            = 324                    ==>        {3246-1, 7} = 3
...
and in general
b_m = 2 + 4∙7+ ... +d_m-1∙7m-1                            ==>        {b_m6-1, 7} = m

But note, that also infinitely many bases of the form

     let x a positive integer such that gcd(x,6) = 1
b₂ = 2 + 4∙7+ x∙72            = 30+x∙49           ==>        {b₂6-1, 7} = 2

give also fermat-quotients of degree 2, so

               {796 - 1, 7} = 2

We can conclude one more result: because we see here a set of infinitely many bases in arithmetic progression, we have even - due to Dirichlet's theorem - a proof, that for each prime p and each degree m we'll find infinitely bases b_m which are prime themselves.

So such a table essentially gives the (p-1)th roots of 1 modulo some pm in the p-adic sense and can in analogy be thought as representing the unit-roots exp(î∙π∙m/(p-1)) in the case of the complex number plane.

 

Note that b=d₀ being a residue (mod p) means, that all bases b gotten by p-adic-expansions beginning at a residue d₀ <>0 (mod p) are also coprime to p and have no (yet recognizable) simple restrictions, so for instance they

  *  can be even,
  *  can be odd,
               and in particular
  *  can be prime themselves.

Sometimes tabulations of Fermat-quotients display only prime bases b - for instance because there is great relevance connected to other exponential diophantine problems like FLT -for instance by the analyses of A. Wieferich (after whom that Wieferich-primes are named) and others. But this is not intended here; here I want to display the general scheme and in that scheme I want to focus to that subset of higher Fermat-quotients where the bases are in the residue classes of the associated primes and thus smaller that that primes.

 

The example computation in the above lines shows immediately, that the displayed bases b = b₁, simply equalling the residue-classes (mod p), are the only ones which are smaller than the prime which defines the congruence. High fermat-quotients are very rare for such cases; two prominent examples are the above mentioned two Wieferich primes p=1093 and p=3511 to base 2 where we have 21092≡1 (mod 10932) and 23510≡1 (mod 35112) or in the new notation {21092-1, 1093} = 2 and {23510-1, 3511} = 2 ; another one is p=11 and the base b=3 where we have {310 - 1, 11} = 2 .

To have such a case is equivalent to have the digit d₁=0 in the table, because that means, according to the example above

b₂ = d₀ + 0∙p1                     = d₀        = b₁        ==>        {b₁p-1 -1 , p} = 2

and the earliest example is that for prime p = 11 and base b₁ = 3

 

p	b	d0	d1	d2	d3	d4	d4	d5	d6	d7	d8	d9	d10	d11
11	3	3	0	1	2	3	6	10	8	7	0	6	3	9

 

(Note that in general zeros in the digits occur randomly scattered over the digits in the table)

The well known Wieferich-primes p=1093 and p=3511 are immediately recognizable in the table by this: they are the only occurences so far, where the root base (= residue d₀) equals 2 (or some small powers of 2) and the digit d₁=0. They are marked orange here:

p	b	d0	d1	d2	d3	d4	d4	d5	d6	d7	d8	d9	d10	d11
1093	2	2	0	974	227	564	478	1009	395	581	1027	834	136	534
3511	2	2	0	102	2379	495	3468	1939	2849	2386	752	3077	803	3499

 

 

 

For even higher Fermat-quotients we can proceed by looking for further consecutive zeros in the digits:

               b₃ = d₀ + 0∙p1+ 0∙p2         = b₀                       ==>        {b₃p-1 - 1, p} = 3
               b_m = d₀ + 0∙p1+ ...+ 0∙pm-1= b₀                      ==>        {b_mp-1 - 1, p} = m
               ...

 

Cases m=3

The only case of the fermat-quotient of order m=3 (where also the base is smaller than the defining prime) occurs with the prime p=113 and the base b₃ = b₁ = d₀ = 68 :

p	b	d0	d1	d2	d3	d4	d4	d5	d6	d7	d8	d9	d10	d11
113	68	68	0	0	92	4	67	65	0	13	7	106	20	16

 

For this I checked the primes up to p=17 389 (which is the 2 000^th prime). In the mathematical-discussionboard MSE (see[2]) one correspondent confirmed that observation and proceeded saying he had found this to be true up to p=104 729, (which is the 10 000'th prime) . In [3] we find that no further example occurs up to p ~ 3.6 ∙ 108

 

Cases m>3

I've not found any higher degree Fermat-quotients for pairs of bases and primes with b<p.

 

Gottfried Helms, 7.12.2013

 

Table 1, b<p:

p-adic representations for bases allowing fermat-quotients up to degree 12, where the high fermat-quotient occurs already for the "root" bases (which are smaller than the defining prime). That is equivalent to say that the p-adic-digit d₁ = 0 .

p	b	d0	d1	d2	d3	d4	d4	d5	d6	d7	d8	d9	d10	d11
11	3	3	0	1	2	3	6	10	8	7	0	6	3	9
11	9	9	0	6	1	9	8	7	1	3	2	0	0	6
29	14	14	0	6	9	19	22	8	24	4	3	19	16	25
37	18	18	0	26	16	3	34	26	35	26	24	18	24	1
43	19	19	0	34	35	10	37	7	26	6	41	34	4	21
59	53	53	0	28	27	40	50	43	2	38	17	21	45	38
71	11	11	0	9	23	19	43	34	58	41	12	49	52	48
71	26	26	0	19	48	7	70	7	9	25	60	45	27	29
79	31	31	0	22	69	9	4	0	69	27	67	11	51	42
97	53	53	0	87	20	43	33	35	56	75	38	17	7	71
103	43	43	0	69	94	23	81	59	78	91	45	75	19	9
109	96	96	0	63	53	40	18	51	44	40	99	57	62	83
113	68	68	0	0	92	4	67	65	0	13	7	106	20	16
127	38	38	0	14	80	89	79	92	87	33	49	34	108	78
127	62	62	0	38	18	74	120	88	26	85	117	22	102	121
131	58	58	0	58	45	43	35	118	39	49	117	20	67	93
131	111	111	0	100	6	68	71	120	87	121	113	124	2	1
137	19	19	0	32	71	56	130	42	88	50	33	95	120	36
151	78	78	0	24	93	15	23	57	16	50	119	20	54	29
163	65	65	0	117	74	136	122	122	90	95	161	79	119	141
163	84	84	0	30	38	143	65	86	83	84	150	160	147	3
181	78	78	0	122	38	129	144	116	13	64	54	10	97	32
191	176	176	0	70	66	133	24	78	95	120	16	148	13	67
197	143	143	0	180	145	6	84	55	185	7	77	174	25	41
199	174	174	0	105	162	83	129	100	104	106	36	10	83	176
211	165	165	0	192	78	98	178	173	190	77	96	101	58	86
211	182	182	0	155	11	37	210	111	113	126	10	199	119	24
223	69	69	0	41	45	55	211	154	104	52	8	140	215	115
229	44	44	0	120	127	35	174	225	143	172	7	161	17	73
229	209	209	0	219	104	7	38	62	94	32	167	209	89	92
233	33	33	0	34	13	124	218	79	170	138	23	22	145	55
241	94	94	0	200	219	44	40	139	133	194	33	79	216	98
257	48	48	0	35	193	232	136	111	42	246	8	240	108	153
263	79	79	0	228	138	101	146	179	235	11	154	235	65	25
269	171	171	0	113	237	101	88	201	166	6	182	69	230	91
269	180	180	0	74	218	219	202	244	259	140	164	82	64	245
269	207	207	0	77	22	243	185	188	90	167	129	189	99	31
281	20	20	0	127	19	267	238	270	101	2	147	168	255	241
283	147	147	0	26	218	169	270	98	171	149	208	28	73	159
293	91	91	0	39	206	171	105	26	97	289	219	38	212	97
307	40	40	0	65	32	263	223	198	59	173	201	300	51	156
313	104	104	0	218	120	107	74	51	239	108	42	125	305	227
313	213	213	0	58	53	223	63	66	120	238	69	293	236	74
331	18	18	0	14	185	36	326	146	77	236	117	314	217	208
331	71	71	0	147	280	26	271	249	28	266	190	81	72	272
331	324	324	0	173	41	188	39	158	186	190	192	261	44	66
347	75	75	0	27	292	130	202	322	311	308	91	124	68	45
347	156	156	0	204	298	48	283	227	292	96	238	158	288	171
349	223	223	0	164	173	80	185	86	312	39	96	135	319	200
349	317	317	0	103	38	68	134	72	47	119	213	260	143	75
353	14	14	0	337	259	116	221	334	144	288	263	294	249	149
353	196	196	0	258	218	347	348	162	301	246	26	14	219	298
359	257	257	0	7	62	98	138	61	136	346	355	202	95	59
359	331	331	0	220	114	233	245	242	343	204	171	157	105	184
367	159	159	0	8	121	119	104	131	157	155	171	361	260	334
367	205	205	0	357	199	82	254	320	156	227	196	115	135	257
373	242	242	0	85	355	365	313	261	262	275	8	304	76	16
379	174	174	0	198	72	194	145	375	164	136	90	120	97	280
397	175	175	0	184	332	45	282	293	362	7	18	355	77	396
401	280	280	0	111	66	332	351	28	254	130	160	12	373	20
419	369	369	0	210	182	164	36	27	57	284	65	109	78	307
421	251	251	0	391	78	129	155	80	102	361	409	247	41	368
433	349	349	0	226	95	120	215	8	94	113	137	206	50	182
439	194	194	0	171	199	141	134	421	15	420	120	321	21	35
449	210	210	0	241	129	137	282	393	314	294	83	246	101	212
461	52	52	0	206	196	154	265	320	98	291	416	232	273	451
463	255	255	0	177	75	422	113	430	83	351	42	169	375	265
463	345	345	0	40	331	112	286	24	426	256	175	307	436	370
487	10	10	0	329	462	103	341	149	54	64	287	349	100	311
487	100	100	0	249	0	258	336	428	281	36	237	54	415	322
487	175	175	0	427	44	471	72	106	273	475	187	406	475	207
487	307	307	0	424	391	356	98	429	194	148	479	236	420	3
499	346	346	0	460	279	269	135	282	230	492	301	219	312	420
509	93	93	0	263	29	65	137	86	45	367	277	294	481	493
509	250	250	0	362	34	224	121	181	23	163	373	211	115	476
521	308	308	0	234	382	126	54	143	270	374	143	129	105	504
523	241	241	0	177	363	517	57	483	214	449	291	455	515	365
547	427	427	0	80	35	1	9	519	12	248	135	52	437	525
557	430	430	0	216	407	507	519	454	443	482	394	423	307	4
563	486	486	0	95	227	453	37	221	314	138	122	430	552	234
571	262	262	0	465	32	170	218	521	229	319	188	359	219	231
571	422	422	0	235	88	536	559	257	229	106	540	414	430	452
571	516	516	0	432	328	306	112	386	407	316	168	298	190	9
577	427	427	0	173	365	178	420	563	350	442	230	281	568	135
599	559	559	0	22	516	267	499	67	485	573	452	454	565	338
599	588	588	0	589	354	135	381	250	166	129	215	282	372	246
601	405	405	0	206	198	25	241	258	187	133	391	525	404	295
607	162	162	0	28	95	427	332	475	143	10	374	525	53	61
607	302	302	0	298	328	238	332	253	427	365	482	386	364	1
617	556	556	0	295	98	413	185	346	272	288	5	439	33	555
619	286	286	0	412	73	19	138	443	323	552	101	519	576	4
631	69	69	0	436	306	380	15	431	309	438	213	508	462	250
631	534	534	0	166	492	272	428	130	153	213	346	378	261	273
641	340	340	0	262	69	543	35	618	472	390	519	613	20	474
647	56	56	0	140	27	415	35	514	425	378	583	89	458	253
647	526	526	0	424	398	143	435	400	187	561	421	578	392	523
653	84	84	0	256	398	21	227	576	63	105	604	360	551	337
653	120	120	0	380	432	266	387	96	541	640	138	311	454	151
653	197	197	0	488	104	542	530	73	535	89	26	154	125	304
653	287	287	0	242	101	139	90	595	75	26	265	559	647	617
653	410	410	0	550	517	360	607	363	295	451	73	367	504	652
659	503	503	0	332	163	234	104	573	123	3	637	498	72	46
661	184	184	0	24	88	510	39	522	472	487	446	32	600	5
673	22	22	0	131	423	461	316	512	460	298	0	549	192	316
673	484	484	0	380	449	457	280	57	208	610	668	555	341	244
701	375	375	0	608	72	366	153	324	202	483	404	383	105	355
727	92	92	0	148	465	314	508	11	31	88	13	164	581	357
739	168	168	0	338	187	646	243	602	64	62	195	681	478	230
743	467	467	0	301	187	637	366	614	176	85	187	552	288	193
761	675	675	0	240	616	172	161	269	43	462	481	759	669	550
769	392	392	0	749	733	564	221	316	264	293	466	340	696	542
773	392	392	0	575	155	542	561	413	446	173	546	81	245	375
797	440	440	0	651	772	713	596	647	154	69	511	416	563	228
797	446	446	0	133	654	744	149	541	441	195	222	529	417	1
809	207	207	0	136	315	270	374	671	514	533	322	556	253	603
823	393	393	0	546	57	76	596	160	414	133	740	627	385	297
827	314	314	0	587	431	569	179	608	385	0	118	506	728	509
827	381	381	0	471	730	400	773	607	299	760	280	656	178	480
829	46	46	0	416	576	9	262	755	19	576	659	306	652	387
829	632	632	0	805	672	312	78	524	275	738	208	683	20	821
839	667	667	0	11	295	797	257	832	360	25	153	134	401	788
857	682	682	0	580	841	472	470	270	553	763	247	42	781	394
859	643	643	0	571	357	16	689	823	4	328	311	716	464	209
863	13	13	0	223	549	234	434	827	732	727	373	463	629	574
863	169	169	0	620	472	597	754	378	551	481	279	579	638	756
863	434	434	0	362	749	6	860	254	560	395	438	178	791	680
877	166	166	0	82	875	839	55	93	562	657	372	541	830	526
883	644	644	0	798	613	531	182	277	738	635	393	228	723	250
883	657	657	0	539	735	114	572	515	757	846	839	13	214	216
883	754	754	0	765	92	747	821	54	410	403	875	655	14	828
887	292	292	0	455	170	748	775	370	490	186	715	498	157	698
907	127	127	0	137	271	198	33	837	240	191	571	828	476	309
907	761	761	0	455	797	276	341	797	449	578	290	269	406	425
907	771	771	0	409	815	892	441	213	535	209	336	728	505	730
911	328	328	0	672	30	58	222	866	865	17	817	391	875	619
919	457	457	0	855	700	794	688	724	363	145	798	323	119	552
941	292	292	0	172	395	299	939	178	489	440	7	689	635	512
947	208	208	0	312	35	913	584	565	6	410	775	879	299	809
953	325	325	0	572	927	616	271	312	900	736	196	723	30	312
967	287	287	0	825	375	427	750	261	32	728	516	219	280	682
971	296	296	0	948	665	509	708	761	603	285	562	329	661	4
977	238	238	0	347	362	124	792	240	200	756	161	243	741	462
977	354	354	0	850	276	686	637	400	930	929	95	105	884	144
983	419	419	0	213	574	546	313	131	413	53	233	388	969	833
991	976	976	0	575	10	287	151	223	170	673	938	644	102	782
997	252	252	0	263	89	35	122	232	177	340	31	656	430	975
1009	990	990	0	124	622	369	650	10	652	877	737	519	691	289
1013	899	899	0	53	939	233	19	435	933	764	290	801	955	370
1019	705	705	0	567	21	532	557	518	854	25	529	594	417	158
1021	551	551	0	260	41	772	221	55	674	42	959	304	485	635
1021	637	637	0	589	121	633	88	773	342	354	91	804	655	333
1031	158	158	0	480	830	356	934	477	622	710	463	770	100	369
1051	806	806	0	473	472	90	578	635	631	429	987	224	39	242
1061	508	508	0	688	141	676	860	1001	444	150	103	850	825	635
1061	917	917	0	927	579	89	823	291	615	182	526	837	78	720
1069	487	487	0	200	281	102	91	1020	732	285	585	688	933	294
1087	617	617	0	69	32	215	116	253	343	137	787	598	20	1069
1087	740	740	0	667	801	656	635	137	1075	554	970	612	603	320
1091	691	691	0	344	654	419	729	142	357	405	134	588	560	604
1093	2	2	0	974	227	564	478	1009	395	581	1027	834	136	534
1093	4	4	0	617	911	22	127	436	652	1082	282	490	680	617
1093	8	8	0	758	548	1017	484	587	858	153	261	275	1033	699
1093	16	16	0	564	734	507	841	302	601	527	806	750	801	954
1093	32	32	0	317	743	855	245	337	933	967	356	787	38	234
1093	64	64	0	105	35	624	721	693	233	641	256	810	308	709
1093	128	128	0	245	446	816	460	1024	275	545	532	401	285	940
1093	256	256	0	560	551	976	443	773	450	275	240	1071	524	75
1093	512	512	0	167	421	1054	720	588	456	112	1072	227	128	1009
1093	1024	1024	0	614	571	46	499	179	113	286	721	989	468	139
1097	425	425	0	26	245	1049	249	341	724	106	41	347	69	1011
1097	579	579	0	763	103	818	258	393	810	261	805	264	120	973
1097	776	776	0	620	916	30	815	652	151	357	698	307	129	138
1103	284	284	0	835	17	453	446	84	354	412	589	730	1074	886
1103	793	793	0	580	1071	306	463	830	699	859	172	222	268	395
1103	1054	1054	0	344	685	433	201	756	483	5	245	675	558	634
1109	76	76	0	939	809	237	480	479	1107	637	129	392	867	156
1109	1082	1082	0	783	361	697	958	48	616	149	794	60	133	1028
1117	1066	1066	0	898	390	571	219	1096	267	878	759	80	868	135
1123	897	897	0	639	1027	895	574	845	571	509	433	73	773	113
1123	1012	1012	0	371	669	175	423	392	604	1042	815	547	645	614
1163	78	78	0	1098	801	411	910	227	499	290	721	643	885	799
1163	170	170	0	428	1101	877	826	57	692	90	574	687	744	605
1163	241	241	0	364	519	987	983	184	662	156	159	92	997	997
1163	618	618	0	941	548	1152	1052	932	236	827	244	465	173	961
1181	874	874	0	304	948	177	215	662	90	55	13	207	287	575
1187	184	184	0	350	618	719	797	67	157	998	345	1017	536	1036
1187	315	315	0	503	1000	945	140	370	541	378	988	948	1132	40
1193	622	622	0	941	639	206	110	543	439	449	389	884	88	140
1201	206	206	0	411	1159	1118	848	842	465	457	1129	292	91	517
1213	178	178	0	20	837	172	364	671	1058	94	1145	1170	110	1039
1217	1188	1188	0	126	385	746	59	346	177	424	356	890	72	454
1223	997	997	0	743	770	969	557	48	55	398	223	771	332	183
1229	821	821	0	1021	765	663	1085	785	661	1217	812	285	612	354
1249	326	326	0	385	232	1234	1185	1083	219	1074	176	99	406	24
1279	683	683	0	170	53	846	1261	481	971	418	535	375	587	287
1283	45	45	0	791	1182	60	228	365	911	1265	1000	455	251	524
1291	62	62	0	17	698	644	798	311	167	837	823	866	1160	319
1291	1148	1148	0	143	119	24	605	147	961	971	1103	732	503	1076
1291	1286	1286	0	1166	961	41	104	765	57	690	797	190	757	476
1297	156	156	0	200	204	688	901	932	41	624	1128	67	85	477
1303	528	528	0	383	754	864	147	1292	651	1285	338	1055	152	1249
1321	403	403	0	899	701	737	674	388	440	885	1064	1095	252	1307
1327	585	585	0	968	642	607	640	378	1080	438	1073	247	1241	448
1327	1149	1149	0	707	865	1142	291	1174	216	1326	266	54	448	133
1361	228	228	0	1135	1151	901	1336	386	44	558	1234	923	1313	838
1367	411	411	0	821	461	915	1129	35	538	716	634	223	1124	1358
1373	884	884	0	291	855	67	378	133	350	331	244	979	1343	1014
1381	429	429	0	250	1342	1333	1196	842	241	273	1002	71	659	223
1381	653	653	0	1157	633	198	1002	1069	345	416	171	120	1354	990
1399	328	328	0	234	1124	836	567	1292	70	462	1223	28	412	264
1399	987	987	0	1081	1095	1373	1158	944	999	786	764	1083	941	1130
1409	1164	1164	0	191	900	260	426	1401	1126	1215	748	1271	31	1220
1429	688	688	0	965	39	739	1081	280	995	462	636	820	913	1174
1429	1401	1401	0	1261	620	1297	515	194	1253	144	945	639	703	502
1439	319	319	0	180	1292	1262	213	1217	1281	930	1111	698	69	362
1447	584	584	0	784	664	929	1198	1312	882	256	972	75	137	341
1453	365	365	0	1325	1105	1307	1371	702	255	1159	856	945	424	960
1453	378	378	0	1242	1441	129	59	662	946	1278	1063	801	1452	117
1459	1082	1082	0	544	379	742	1112	1340	292	1061	1180	1124	698	876
1471	1195	1195	0	1397	299	935	820	319	95	1171	855	395	152	920
1483	421	421	0	1302	215	604	640	1343	1203	990	1062	138	975	162
1483	1061	1061	0	264	408	907	1421	487	226	897	1365	1434	749	1481
1489	1211	1211	0	556	88	270	1432	1470	412	126	518	949	6	582
1493	164	164	0	704	518	340	810	833	1474	846	704	592	817	152
1493	488	488	0	1016	1446	1266	958	950	854	799	1103	245	633	690
1499	941	941	0	981	1381	1464	692	260	963	1104	557	22	1084	634
1499	1172	1172	0	1375	855	1059	635	118	1181	1408	469	1232	1096	1
1511	934	934	0	953	1124	161	281	787	1217	569	997	23	524	976
1523	1032	1032	0	1478	524	1315	132	326	23	62	1499	1183	1429	309
1523	1246	1246	0	646	83	120	916	960	240	680	604	78	111	1322
1531	472	472	0	574	1110	1386	467	1395	180	765	1143	1155	164	1241
1531	1238	1238	0	1323	1331	401	904	724	1315	853	926	280	731	835
1549	855	855	0	1003	313	1545	254	1010	256	1073	197	1028	586	448
1549	1069	1069	0	1248	215	543	885	1211	194	1459	585	265	1319	444
1553	392	392	0	588	1267	615	1068	1300	144	1149	1018	230	347	52
1553	568	568	0	321	774	1018	641	1413	1423	1035	1369	225	199	875
1559	1454	1454	0	56	668	1379	798	1209	98	538	701	963	826	19
1571	1265	1265	0	116	859	989	111	649	1	538	780	6	1080	201
1579	603	603	0	709	1177	1270	1286	1092	1305	592	55	1000	1327	1496
1579	906	906	0	201	345	1225	73	749	532	134	1381	1115	754	24
1579	1235	1235	0	619	916	1548	1353	1051	681	1411	487	199	1283	289
1597	453	453	0	473	345	564	567	998	205	1548	1152	1016	148	499
1601	1420	1420	0	1	1562	600	336	214	1199	81	638	1296	384	139
1607	874	874	0	669	1209	508	82	402	146	1188	450	643	441	92
1607	1253	1253	0	64	1021	489	1135	786	892	1418	1564	1437	326	1497
1613	35	35	0	359	1194	748	952	514	222	858	510	851	401	1067
1613	1225	1225	0	935	1332	636	1408	745	906	434	185	333	645	329
1619	371	371	0	37	1339	439	801	1251	1424	19	782	729	1067	226
1619	536	536	0	22	116	95	1380	1340	394	1364	1198	1129	710	24
1621	558	558	0	256	278	405	1138	1452	564	925	313	1599	732	636
1621	746	746	0	1560	1116	604	652	1436	1356	39	1370	1253	486	1301
1627	923	923	0	400	1481	342	1080	1053	842	1070	1303	1240	146	1598
1657	427	427	0	1511	620	1647	1265	1649	510	96	739	1301	333	648
1657	1481	1481	0	1616	369	1222	183	464	566	254	292	475	739	177
1663	709	709	0	1426	1234	998	320	1239	271	173	352	16	68	1158
1667	463	463	0	59	1534	373	1039	315	1371	261	968	884	1362	295
1669	221	221	0