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.
Some preliminary definitions:
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 bm 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:
bm = "dm-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
bm = 2 + 4∙7+ ... +dm-1∙7m-1 ==> {bm6-1, 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 bm 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.
(end of preliminary definitions)
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.
Base b smaller than prime p
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 |
Higher degrees of Fermatquotiens
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
bm = d0
+ 0∙p1+
...+ 0∙pm-1= b0 ==>
{bmp-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 000th 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 |
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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 |