Science Forums 2

CraigD · 02-11-2009

A slightly smarter checking program found
$72^5 = 67^5 +47^5 +46^5 +43^5 +19^5$ , $n=5$ , $m=5$ .

After an hour or two, the best it found for $n$ = 4, 6, and 7 are
$280^4 = 224^4 +208^4 +176^4 +168^4 +32^4$ , $m=5$
$72^6 = 71^6 +40^6 +37^6 +37^6 +35^6 +23^6 +5^6 +5^6 +1^6$ , $m=9$
$54^7 = 49^7 +47^7 +39^7 +27^7 +24^7 +21^7 +5^7 +4^7 +3^7 +2^7 +1^7$ , $m=11$

The program outputs $a_i$ m-tuples for a given $n$ for increasing values of $b$ , avoiding increases in $m$ . Here’s its MUMPS code:

Code:

k A1,B s (B,A1,B1)=0 ;XFMTG2(1): initialize
s B=B+1,R=B**N,B(R)=B zt:R+1=R  k A s A=0 f  s A=A+1,F=$g(A(A)),R=R+F s:'F F=$s($g(A(A-1),R)<R:A(A-1)+1,1:A>1+R) s F=$o(B(F),-1),R=R-F s:$s(F:R/F+A>A1&A1,1:0) R=R+F,F=0 s A(A)=F s:'F A=A-2 q:A<0  i F,'R k A1 m A1=A s B1=B ;XFMTG2(2): check next B, update A1 least A count
w !,N," ",B," ",B1," ",A1,":" f I=1:1:A1 w " ",B(A1(I)) ;XFMTG2(3): display

s N=3 x XFMTG2(1) f  x XFMTG2(2),XFMTG2(3) r R

And its first 10 lines of output for $n=3$ :

Code:

3 1 0 0:
3 2 2 8: 1 1 1 1 1 1 1 1
3 3 3 6: 2 2 2 1 1 1
3 4 4 5: 3 3 2 1 1
3 5 4 5: 3 3 2 1 1
3 6 6 3: 5 4 3
3 7 6 3: 5 4 3
3 8 6 3: 5 4 3
3 9 9 3: 8 6 1
3 10 9 3: 8 6 1

I’m hunching toward conjecturing there’re $b$ and $a_i$ m-tuples such that $m = n$ for all $n$ , but am far from either an efficient program for finding examples, or ideas for proofs.

Science Forums 2

Hypography?

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