Determine all possible
rational u satisfying:
u = [u]*{u}, such that:
5*{u} - [u]/4 is an integer.
Note: [x] is the greatest integer ≤ x, and {x} = x - [x]
If u is positive, [u]*{u} must be less than u, and so can't be equal, so there are no positive values of u that work.
Zero works trivially, and so is one of the values of u.
The following UBASIC program tests all negative rational u where the sum of the absolute values of the numerator and denominator is 20,000 or less:
10 for T=1 to 20000
20 for N=1 to T-1
30 D=T-N
35 if gcd(N,D)=1 then
40 :U=-N//D
50 :Iu=int(U)
60 :Fu=U-Iu
70 :if U=Iu*Fu then
80 :if 5*Fu-Iu//4=int(5*Fu-Iu//4) then
90 :print U
100 next
110 next
It finds only -16/5, where
-16/5 = -4 * 4/5 and
5 * 4/5 - (-4/4) = 4 + 1 = 5, an integer.
Now someone can find a proof that only 0 and -16/5 work.
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Posted by Charlie
on 2008-03-08 12:50:42 |
computer search (spoiler; no proof)
