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Power Difference (Posted on 2011-11-06) Difficulty: 3 of 5
Determine, with proof, the minimum positive value of 33m - 7n, where each of m and n is a positive integer.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer exploration may lead to ideas for a proof | Comment 1 of 5

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

  Posted by Charlie on 2011-11-06 12:01:24
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