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 Greatest Root Power (Posted on 2011-01-12)
Determine all possible pair(s) (M, N) of positive integers such that:

N[√N] = MM-1

Prove that these are the only possible pair(s) that exist.

Note: [x] denotes the greatest integer ≤ x.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer exploration--no proof (spoiler) Comment 1 of 1
`10 for t=1 to 100000020      for n=1 to t-130        m=t-n40        lhs=n^int(sqrt(n)):rhs=m^(m-1)50        if lhs=rhs then ?n,m,lhs60      next n70 next t`
` n       m     lhs (=rhs) 1       1       1 2       2       2 8       4       64Overflow in 40?t 888OK`

shows that for n+m < 888 there are only the three solutions:

(1,1)
(2,2)
(4,8)

A more sensibly written program (one that allows higher values) verifies no more solutions up to n = 48888, the point at which it was stopped:

`10 for n=1 to 99999920    v=n^int(sqrt(n))30    v2=0:m=040    while v2<v50       m=m+1:v2=m^(m-1)60    wend70    if v2=v then ?m,n,v80 nextrun`
` m       n      value 1       1       1 2       2       2 4       8       64Break in 50?n 34644OKcontBreak in 50?n 48888OK`

 Posted by Charlie on 2011-01-12 13:09:55

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