Diophantine equation with factorial
all at abouthugo.de
all at abouthugo.de
Tue Jun 24 23:28:02 CEST 2003
Richard Guy <rkg at cpsc.ucalgary.ca> schrieb am 24.06.2003, 16:46:10:
> Hand-waving ... Most of the primes dividing
> n! do so only to the first power. Half of
> these don't divide x^2 - xy + y^2 and so
> would have to divide x + y. But x + y is small
> compared with the quadratic ... Maybe you
> are allowing y to be negative, in which case
> this `proof' fails ?? R.
>
> On Wed, 25 Jun 2003, Don McDonald wrote:
>
> > In message you write:
> > Richard Guy:
> >
> > > Where angels fear to tread ?
> > (Fools rush in?? )
> > >
> > > x^3 + y^3 = (x + y)(x^2 - xy + y^2) and all squarefree
> > > factors of the latter (paren) are 3 or of form 6k + 1. R.
> > >
> > Sorry, I do not follow. Please Richard.
> > So intermediate factors ..., 6k+5,... may be very thin, sparse?
> >
> > Could you kindly say some more please? Thank you.
Richard, Don, SeqFans,
I also had (and still have) problems to find a
complete proof of "x^3+y^3=n! has only one solution."
I even think (guess after some computer seach),
that there is no solution if we admit negative y.
The easy cases are those when 6k+1 contains
a large prime factor p. Then n! has to contain
(p-1)*(p-2)*...and x+y is far too small to
deliver (p-1)*(p-2). To get an idea about the
other cases I wrote a little program:
(sorry, but for me nothing is obvious, until I can
write a program that spits out results)
C Try to understand why x^3+y^3=n!
C has only one solution x=y=1,
C get idea for proof
C Hugo Pfoertner, http://www.pfoertner.org
C Input from Richard Guy used
C
IMPLICIT INTEGER (A-Z)
C
C LOOPS OVER X AND Y
DO 10 X = 1, 12
X2 = X * X
X3 = X2 * X
C
DO 20 Y = 1, X-1
Y2 = Y * Y
Y3 = Y2 * Y
C X^3+Y^3 = (X+Y)*(X^2-X*Y+Y^2) = YPY*PAR = SUM
XPY = X + Y
PAR = X2 - X*Y + Y2
SUM = X3 + Y3
WRITE (*,1000) X, Y, XPY, PAR, SUM
1000 FORMAT ( I3, '^3+', I2, '^3=', I2, '*(', I3, ')=', I4 )
C
C TRY IF PAR IS OF THE FORM 6*K+1
C
IF ( MOD ( PAR-1,6 ) .EQ. 0 ) THEN
WRITE (*,1001) PAR, 1, (PAR-1)/6, 1, PAR
1001 FORMAT ( ' PAR=', I3, '=',
& I3, '*(6*', I2, '+1)', '=', I3, '*', I3 )
GOTO 20
ENDIF
C
C SEARCH PRODUCT OF SMALL FACTORS F
C
F = 1
PACOPY = PAR
C
C CHECK FOR FACTORS 3 IN PAR
C
30 CONTINUE
IF ( MOD(PACOPY,3) .EQ. 0 ) THEN
F = 3 * F
PACOPY = PACOPY/3
GOTO 30
ENDIF
C
C CHECK FOR FACTORS 2 IN PAR
C
40 CONTINUE
IF ( MOD(PACOPY,2) .EQ. 0 ) THEN
F = 2 * F
PACOPY = PACOPY/2
GOTO 40
ENDIF
C
C AFTER DIVIDING OFF FACTORS 2 AND 3 CHECK
C AGAIN FOR FORM 6*K+1
C
IF ( MOD(PACOPY-1,6) .EQ. 0 ) THEN
C IF THIS TEST IS NOT PASSED, PAR WILL NOT BE PRINTED
WRITE (*,1001) PAR, F, (PACOPY-1)/6, F, PACOPY
ENDIF
C END LOOPS OVER X AND Y
20 CONTINUE
10 CONTINUE
C END OF PROGRAM
END
which produces the following table:
(line limit exceeded? here is my
---- petition for mercy to Olivier Gerard ----)
2^3+ 1^3= 3*( 3)= 9
PAR= 3= 3*(6* 0+1)= 3* 1
3^3+ 1^3= 4*( 7)= 28
PAR= 7= 1*(6* 1+1)= 1* 7
3^3+ 2^3= 5*( 7)= 35
PAR= 7= 1*(6* 1+1)= 1* 7
4^3+ 1^3= 5*( 13)= 65
PAR= 13= 1*(6* 2+1)= 1* 13
4^3+ 2^3= 6*( 12)= 72
PAR= 12= 12*(6* 0+1)= 12* 1
4^3+ 3^3= 7*( 13)= 91
PAR= 13= 1*(6* 2+1)= 1* 13
5^3+ 1^3= 6*( 21)= 126
PAR= 21= 3*(6* 1+1)= 3* 7
5^3+ 2^3= 7*( 19)= 133
PAR= 19= 1*(6* 3+1)= 1* 19
5^3+ 3^3= 8*( 19)= 152
PAR= 19= 1*(6* 3+1)= 1* 19
5^3+ 4^3= 9*( 21)= 189
PAR= 21= 3*(6* 1+1)= 3* 7
6^3+ 1^3= 7*( 31)= 217
PAR= 31= 1*(6* 5+1)= 1* 31
6^3+ 2^3= 8*( 28)= 224
PAR= 28= 4*(6* 1+1)= 4* 7
6^3+ 3^3= 9*( 27)= 243
PAR= 27= 27*(6* 0+1)= 27* 1
6^3+ 4^3=10*( 28)= 280
PAR= 28= 4*(6* 1+1)= 4* 7
6^3+ 5^3=11*( 31)= 341
PAR= 31= 1*(6* 5+1)= 1* 31
7^3+ 1^3= 8*( 43)= 344
PAR= 43= 1*(6* 7+1)= 1* 43
7^3+ 2^3= 9*( 39)= 351
PAR= 39= 3*(6* 2+1)= 3* 13
7^3+ 3^3=10*( 37)= 370
PAR= 37= 1*(6* 6+1)= 1* 37
7^3+ 4^3=11*( 37)= 407
PAR= 37= 1*(6* 6+1)= 1* 37
7^3+ 5^3=12*( 39)= 468
PAR= 39= 3*(6* 2+1)= 3* 13
7^3+ 6^3=13*( 43)= 559
PAR= 43= 1*(6* 7+1)= 1* 43
8^3+ 1^3= 9*( 57)= 513
PAR= 57= 3*(6* 3+1)= 3* 19
8^3+ 2^3=10*( 52)= 520
PAR= 52= 4*(6* 2+1)= 4* 13
8^3+ 3^3=11*( 49)= 539
PAR= 49= 1*(6* 8+1)= 1* 49
8^3+ 4^3=12*( 48)= 576
PAR= 48= 48*(6* 0+1)= 48* 1
8^3+ 5^3=13*( 49)= 637
PAR= 49= 1*(6* 8+1)= 1* 49
8^3+ 6^3=14*( 52)= 728
PAR= 52= 4*(6* 2+1)= 4* 13
8^3+ 7^3=15*( 57)= 855
PAR= 57= 3*(6* 3+1)= 3* 19
9^3+ 1^3=10*( 73)= 730
PAR= 73= 1*(6*12+1)= 1* 73
9^3+ 2^3=11*( 67)= 737
PAR= 67= 1*(6*11+1)= 1* 67
9^3+ 3^3=12*( 63)= 756
PAR= 63= 9*(6* 1+1)= 9* 7
9^3+ 4^3=13*( 61)= 793
PAR= 61= 1*(6*10+1)= 1* 61
9^3+ 5^3=14*( 61)= 854
PAR= 61= 1*(6*10+1)= 1* 61
9^3+ 6^3=15*( 63)= 945
PAR= 63= 9*(6* 1+1)= 9* 7
9^3+ 7^3=16*( 67)=1072
PAR= 67= 1*(6*11+1)= 1* 67
9^3+ 8^3=17*( 73)=1241
PAR= 73= 1*(6*12+1)= 1* 73
10^3+ 1^3=11*( 91)=1001
PAR= 91= 1*(6*15+1)= 1* 91
10^3+ 2^3=12*( 84)=1008
PAR= 84= 12*(6* 1+1)= 12* 7
10^3+ 3^3=13*( 79)=1027
PAR= 79= 1*(6*13+1)= 1* 79
10^3+ 4^3=14*( 76)=1064
PAR= 76= 4*(6* 3+1)= 4* 19
10^3+ 5^3=15*( 75)=1125
PAR= 75= 3*(6* 4+1)= 3* 25
10^3+ 6^3=16*( 76)=1216
PAR= 76= 4*(6* 3+1)= 4* 19
10^3+ 7^3=17*( 79)=1343
PAR= 79= 1*(6*13+1)= 1* 79
10^3+ 8^3=18*( 84)=1512
PAR= 84= 12*(6* 1+1)= 12* 7
10^3+ 9^3=19*( 91)=1729
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 1^3=12*(111)=1332
PAR=111= 3*(6* 6+1)= 3* 37
11^3+ 2^3=13*(103)=1339
PAR=103= 1*(6*17+1)= 1*103
11^3+ 3^3=14*( 97)=1358
PAR= 97= 1*(6*16+1)= 1* 97
11^3+ 4^3=15*( 93)=1395
PAR= 93= 3*(6* 5+1)= 3* 31
11^3+ 5^3=16*( 91)=1456
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 6^3=17*( 91)=1547
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 7^3=18*( 93)=1674
PAR= 93= 3*(6* 5+1)= 3* 31
11^3+ 8^3=19*( 97)=1843
PAR= 97= 1*(6*16+1)= 1* 97
11^3+ 9^3=20*(103)=2060
PAR=103= 1*(6*17+1)= 1*103
11^3+10^3=21*(111)=2331
PAR=111= 3*(6* 6+1)= 3* 37
12^3+ 1^3=13*(133)=1729
PAR=133= 1*(6*22+1)= 1*133
12^3+ 2^3=14*(124)=1736
PAR=124= 4*(6* 5+1)= 4* 31
12^3+ 3^3=15*(117)=1755
PAR=117= 9*(6* 2+1)= 9* 13
12^3+ 4^3=16*(112)=1792
PAR=112= 16*(6* 1+1)= 16* 7
12^3+ 5^3=17*(109)=1853
PAR=109= 1*(6*18+1)= 1*109
12^3+ 6^3=18*(108)=1944
PAR=108=108*(6* 0+1)=108* 1
12^3+ 7^3=19*(109)=2071
PAR=109= 1*(6*18+1)= 1*109
12^3+ 8^3=20*(112)=2240
PAR=112= 16*(6* 1+1)= 16* 7
12^3+ 9^3=21*(117)=2457
PAR=117= 9*(6* 2+1)= 9* 13
12^3+10^3=22*(124)=2728
PAR=124= 4*(6* 5+1)= 4* 31
12^3+11^3=23*(133)=3059
PAR=133= 1*(6*22+1)= 1*133
Does that help to find a proof?
Hugo 2^3+ 1^3= 3*( 3)= 9
PAR= 3= 3*(6* 0+1)= 3* 1
3^3+ 1^3= 4*( 7)= 28
PAR= 7= 1*(6* 1+1)= 1* 7
3^3+ 2^3= 5*( 7)= 35
PAR= 7= 1*(6* 1+1)= 1* 7
4^3+ 1^3= 5*( 13)= 65
PAR= 13= 1*(6* 2+1)= 1* 13
4^3+ 2^3= 6*( 12)= 72
PAR= 12= 12*(6* 0+1)= 12* 1
4^3+ 3^3= 7*( 13)= 91
PAR= 13= 1*(6* 2+1)= 1* 13
5^3+ 1^3= 6*( 21)= 126
PAR= 21= 3*(6* 1+1)= 3* 7
5^3+ 2^3= 7*( 19)= 133
PAR= 19= 1*(6* 3+1)= 1* 19
5^3+ 3^3= 8*( 19)= 152
PAR= 19= 1*(6* 3+1)= 1* 19
5^3+ 4^3= 9*( 21)= 189
PAR= 21= 3*(6* 1+1)= 3* 7
6^3+ 1^3= 7*( 31)= 217
PAR= 31= 1*(6* 5+1)= 1* 31
6^3+ 2^3= 8*( 28)= 224
PAR= 28= 4*(6* 1+1)= 4* 7
6^3+ 3^3= 9*( 27)= 243
PAR= 27= 27*(6* 0+1)= 27* 1
6^3+ 4^3=10*( 28)= 280
PAR= 28= 4*(6* 1+1)= 4* 7
6^3+ 5^3=11*( 31)= 341
PAR= 31= 1*(6* 5+1)= 1* 31
7^3+ 1^3= 8*( 43)= 344
PAR= 43= 1*(6* 7+1)= 1* 43
7^3+ 2^3= 9*( 39)= 351
PAR= 39= 3*(6* 2+1)= 3* 13
7^3+ 3^3=10*( 37)= 370
PAR= 37= 1*(6* 6+1)= 1* 37
7^3+ 4^3=11*( 37)= 407
PAR= 37= 1*(6* 6+1)= 1* 37
7^3+ 5^3=12*( 39)= 468
PAR= 39= 3*(6* 2+1)= 3* 13
7^3+ 6^3=13*( 43)= 559
PAR= 43= 1*(6* 7+1)= 1* 43
8^3+ 1^3= 9*( 57)= 513
PAR= 57= 3*(6* 3+1)= 3* 19
8^3+ 2^3=10*( 52)= 520
PAR= 52= 4*(6* 2+1)= 4* 13
8^3+ 3^3=11*( 49)= 539
PAR= 49= 1*(6* 8+1)= 1* 49
8^3+ 4^3=12*( 48)= 576
PAR= 48= 48*(6* 0+1)= 48* 1
8^3+ 5^3=13*( 49)= 637
PAR= 49= 1*(6* 8+1)= 1* 49
8^3+ 6^3=14*( 52)= 728
PAR= 52= 4*(6* 2+1)= 4* 13
8^3+ 7^3=15*( 57)= 855
PAR= 57= 3*(6* 3+1)= 3* 19
9^3+ 1^3=10*( 73)= 730
PAR= 73= 1*(6*12+1)= 1* 73
9^3+ 2^3=11*( 67)= 737
PAR= 67= 1*(6*11+1)= 1* 67
9^3+ 3^3=12*( 63)= 756
PAR= 63= 9*(6* 1+1)= 9* 7
9^3+ 4^3=13*( 61)= 793
PAR= 61= 1*(6*10+1)= 1* 61
9^3+ 5^3=14*( 61)= 854
PAR= 61= 1*(6*10+1)= 1* 61
9^3+ 6^3=15*( 63)= 945
PAR= 63= 9*(6* 1+1)= 9* 7
9^3+ 7^3=16*( 67)=1072
PAR= 67= 1*(6*11+1)= 1* 67
9^3+ 8^3=17*( 73)=1241
PAR= 73= 1*(6*12+1)= 1* 73
10^3+ 1^3=11*( 91)=1001
PAR= 91= 1*(6*15+1)= 1* 91
10^3+ 2^3=12*( 84)=1008
PAR= 84= 12*(6* 1+1)= 12* 7
10^3+ 3^3=13*( 79)=1027
PAR= 79= 1*(6*13+1)= 1* 79
10^3+ 4^3=14*( 76)=1064
PAR= 76= 4*(6* 3+1)= 4* 19
10^3+ 5^3=15*( 75)=1125
PAR= 75= 3*(6* 4+1)= 3* 25
10^3+ 6^3=16*( 76)=1216
PAR= 76= 4*(6* 3+1)= 4* 19
10^3+ 7^3=17*( 79)=1343
PAR= 79= 1*(6*13+1)= 1* 79
10^3+ 8^3=18*( 84)=1512
PAR= 84= 12*(6* 1+1)= 12* 7
10^3+ 9^3=19*( 91)=1729
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 1^3=12*(111)=1332
PAR=111= 3*(6* 6+1)= 3* 37
11^3+ 2^3=13*(103)=1339
PAR=103= 1*(6*17+1)= 1*103
11^3+ 3^3=14*( 97)=1358
PAR= 97= 1*(6*16+1)= 1* 97
11^3+ 4^3=15*( 93)=1395
PAR= 93= 3*(6* 5+1)= 3* 31
11^3+ 5^3=16*( 91)=1456
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 6^3=17*( 91)=1547
PAR= 91= 1*(6*15+1)= 1* 91
11^3+ 7^3=18*( 93)=1674
PAR= 93= 3*(6* 5+1)= 3* 31
11^3+ 8^3=19*( 97)=1843
PAR= 97= 1*(6*16+1)= 1* 97
11^3+ 9^3=20*(103)=2060
PAR=103= 1*(6*17+1)= 1*103
11^3+10^3=21*(111)=2331
PAR=111= 3*(6* 6+1)= 3* 37
12^3+ 1^3=13*(133)=1729
PAR=133= 1*(6*22+1)= 1*133
12^3+ 2^3=14*(124)=1736
PAR=124= 4*(6* 5+1)= 4* 31
12^3+ 3^3=15*(117)=1755
PAR=117= 9*(6* 2+1)= 9* 13
12^3+ 4^3=16*(112)=1792
PAR=112= 16*(6* 1+1)= 16* 7
12^3+ 5^3=17*(109)=1853
PAR=109= 1*(6*18+1)= 1*109
12^3+ 6^3=18*(108)=1944
PAR=108=108*(6* 0+1)=108* 1
12^3+ 7^3=19*(109)=2071
PAR=109= 1*(6*18+1)= 1*109
12^3+ 8^3=20*(112)=2240
PAR=112= 16*(6* 1+1)= 16* 7
12^3+ 9^3=21*(117)=2457
PAR=117= 9*(6* 2+1)= 9* 13
12^3+10^3=22*(124)=2728
PAR=124= 4*(6* 5+1)= 4* 31
12^3+11^3=23*(133)=3059
PAR=133= 1*(6*22+1)= 1*133
Does this help to get a proof?
Hugo
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