a b a^2b+a+b ab^2+b+7 a/7 b/7
11 1 133 19 11/7 1/7
7 7 357 357 1 1
28 14 11018 5509 4 2
49 1 2451 57 7 1/7
63 21 83433 27811 9 3
112 28 351372 87843 16 4
175 35 1072085 214417 25 5
252 42 2667462 444577 36 6
343 49 5765193 823599 49 7
448 56 11239928 1404991 64 8
567 63 20254437 2250493 81 9
700 70 34300770 3430077 100 10
847 77 55241417 5021947 121 11
1008 84 85350468 7112539 144 12
1183 91 127354773 9796521 169 13
1372 98 184475102 13176793 196 14
1575 105 260467305 17364487 225 15
1792 112 359663472 22478967 256 16
2023 119 487013093 28647829 289 17
2268 126 648124218 36006901 324 18
2527 133 849304617 44700243 361 19
2800 140 1097602940 54880147 400 20
3087 147 1400849877 66707137 441 21
3388 154 1767699318 80349969 484 22
3703 161 2207669513 95985631 529 23
4032 168 2731184232 113799343 576 24
4375 175 3349613925 133984557 625 25
4732 182 4075316882 156742957 676 26
5103 189 4921680393 182284459 729 27
5488 196 5903161908 210827211 784 28
5887 203 7035330197 242597593 841 29
6300 210 8334906510 277830217 900 30
6727 217 9819805737 316767927 961 31
7168 224 11509177568 359661799 1024 32
7623 231 13423447653 406771141 1089 33
8092 238 15584358762 458363493 1156 34
8575 245 18015011945 514714627 1225 35
9072 252 20739907692 576108547 1296 36
9583 259 23784987093 642837489 1369 37
10108 266 27177672998 715201921 1444 38
10647 273 30946911177 793510543 1521 39
11200 280 35123211480 878080287 1600 40
11767 287 39738688997 969236317 1681 41
12348 294 44827105218 1067312029 1764 42
12943 301 50423909193 1172649051 1849 43
10 for T=1 to 1000000
20 for A=1 to T-1
30 B=T-A
40 R=(A*A*B+A+B)@(A*B*B+B+7)
50 if R=0 then print A;B,A*A*B+A+B;A*B*B+B+7;A*B*B+B+7,A//7;B//7
60 next
70 next
was stopped when the DOS screen became full.
Aside from (11,1) and (49,1) any (7x^2,7x) seems to work.
Doing the division (7^3*x^5 + 7*x^2 + 7*x) / (7^3*x^4 + 7*x + 7) gives x with no remainder, verifying the sufficiency of the latter part of the solution.
Probably there are no other sporadics like (11,1) and (49,1).
Edited on February 15, 2009, 2:20 pm
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Posted by Charlie
on 2009-02-15 14:19:34 |
partial list -- possible spoiler.
