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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 actually... | Comment 2 of 5 |
(In reply to computer exploration may lead to ideas for a proof by Charlie)

... we want the closest positive, so we should seek only positives, and truncate the value of n for a given m, rather than round it. The minimum then would be the 26 from m=1, n=1:

 

   10   for M=1 to 40
   20    P33=33^M
   30    N=int(log(P33)/log(7))
   40    P7=7^N
   50    print M,N,P33-P7
   60   next
run
 1       1       26
 2       3       746
 3       5       19130
 4       7       362378
 5       8       33370592
 6       10      1008992720
 7       12      28777155776
 8       14      728185545392
 9       16      13178553832352
 10      17      1298948471277242
 11      19      39143211328353674
 12      21      1109343650869700954
 13      23      27671606653367587370
 14      25      475263062119835721722
 15      26      50551465161217666237808
 16      28      1517998664917818916957280
 17      30      42734171357572184883961424
 18      32      1049598210148805971764915008
 19      34      16965898147007866727357408048
 20      35      1966915495838014505395745647658
 21      37      58847112286403842171136124357626
 22      39      1644960850715045602808241485062346
 23      41      39731009191518726639548688688831130
 24      43      598041058098306844702065991138339178
 25      44      76514528693030837452010296279164833792
 26      46      2280392236758936416816096804904010292720
 27      48      63268170517601922019019288918719307761376
 28      50      1500595735604157419567953157805198593104592
 29      52      20744218592578600499817968758457163984431552
 30      53      2975803677887896891288230122914819682432332442
 31      55      88331545216251599551859495547142813891576116074
 32      57      2431312160587901890039410526112447312616159959354
 33      59      56535488553529118507874858841490749571846536975370
 34      61      704478297718750361452011604278414160391085377704922
 35      62      115707762479330214416554634913891070615488900984414608
 36      64      3420067022998032357333707405199067592921545065014257280
 37      66      93346043956761696589795162570561681434321142439370675824
 38      68      2124127228266640798554598456579165579860195309900341395808
 39      70      23237879639780882095778295562893338669238728087365611648848
 40      71      4497943669969348400586363738848928041326890004103462917032458

  Posted by Charlie on 2011-11-06 12:05:50
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