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 Power Difference (Posted on 2011-11-06)
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 No Rating

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 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   nextrun 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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