Based on the below, which finds the optimum n for given values of m, it would seem the answer is 19,130 for m=3, n=5, as the differences all seem to be of the order of magnitude of the numbers being subtracted.
10 for M=1 to 40
20 P33=33^M
30 N=int(log(P33)/log(7)+0.5)
40 P7=7^N
50 print M,N,P33-P7
60 next
run
1 2 -16
2 4 -1312
3 5 19130
4 7 362378
5 9 -1218214
6 11 -685858774
7 13 -54270567430
8 14 728185545392
9 16 13178553832352
10 18 -96834612646000
11 20 -29250159783885184
12 22 -2241931533630003088
13 23 27671606653367587370
14 25 475263062119835721722
15 27 -5773416864668859596086
16 29 -1241920554350620848903526
17 31 -92501870386581363643218070
18 32 1049598210148805971764915008
19 34 16965898147007866727357408048
20 36 -305996657755974184700560108000
21 38 -52525583239701603643582857669616
22 40 -3812301230064121242112988634272512
23 41 39731009191518726639548688688831130
24 43 598041058098306844702065991138339178
25 45 -15205675098624619710580818336496124614
26 47 -2213897749032180984150867811263376669174
27 49 -156952038786162830628361977273482653371430
28 50 1500595735604157419567953157805198593104592
29 52 20744218592578600499817968758457163984431552
30 54 -725437379880477306456306817977518678326738000
31 56 -93029266614398736137622814556581765785618335584
32 57 2431312160587901890039410526112447312616159959354
33 59 56535488553529118507874858841490749571846536975370
34 61 704478297718750361452011604278414160391085377704922
35 63 -33650664578119054988168695195860580905606849317369686
36 65 -3898495902816981843497735770178763331612146699773173126
37 66 93346043956761696589795162570561681434321142439370675824
38 68 2124127228266640798554598456579165579860195309900341395808
39 70 23237879639780882095778295562893338669238728087365611648848
40 72 -1529207597645125839408965468131256971756360112944871881816000
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Posted by Charlie
on 2011-11-06 12:01:24 |
computer exploration may lead to ideas for a proof
