Phil Carmody on Thu, 3 Jul 2003 08:39:26 -0700 (PDT)

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isprime(,2) (again, it seems)

Looking back through the dev archives (to Jan this year, but not further
back, apologies if this has been mentioned before then), it appears that
the APRCL in 2.2.5 has a few problems when trying to prove some numbers 
of the form Phi(a,b). 

In particular there are two different failure modes.

1) ***   non invertible matrix in gauss,
   e.g. gp > isprime(subst(polcyclo(16),x,16),2)

2) Digagreement with isprime(,1)
(18:33) gp > isprime(subst(polcyclo(36),x,125),1)
%6 = 
[2 11 1]
[5167 2 1]

(18:33) gp > isprime(subst(polcyclo(36),x,125),2)
%7 = 0

Here are the values for 2<=a<=150, 2<=b<=200. As you can see there's a bit
of a pattern to them. Ones with odd b I think are failure (b) above.

Phi(16,16) Phi(16,144)

Phi(32,4) Phi(32,12) Phi(32,16) Phi(32,20) Phi(32,28) Phi(32,36) Phi(32,44) 
Phi(32,48) Phi(32,52) Phi(32,64) Phi(32,76) Phi(32,80) Phi(32,84) Phi(32,92)
Phi(32,100) Phi(32,108) Phi(32,112) Phi(32,116) Phi(32,132) Phi(32,140)
Phi(32,148) Phi(32,156) Phi(32,164) Phi(32,172) Phi(32,176) Phi(32,188)
Phi(32,192) Phi(32,196) Phi(32,198) 


Phi(48,16) Phi(48,80) Phi(48,96) Phi(48,112) 


Phi(64,2) Phi(64,4) Phi(64,6) Phi(64,8) Phi(64,10) Phi(64,14) Phi(64,16)
Phi(64,18) Phi(64,24) Phi(64,26) Phi(64,28) Phi(64,30) Phi(64,32) Phi(64,34)
Phi(64,36) Phi(64,38) Phi(64,40) Phi(64,48) Phi(64,52) Phi(64,54) Phi(64,56)
Phi(64,58) Phi(64,62) Phi(64,64) Phi(64,66) Phi(64,72) Phi(64,82) Phi(64,86)
Phi(64,96) Phi(64,98) Phi(64,100) Phi(64,102) Phi(64,108) Phi(64,110) 
Phi(64,112) Phi(64,114) Phi(64,116) Phi(64,118) Phi(64,120) Phi(64,124)
Phi(64,126) Phi(64,128) Phi(64,130) Phi(64,132) Phi(64,136) Phi(64,138)
Phi(64,142) Phi(64,144) Phi(64,148) Phi(64,150) Phi(64,152) Phi(64,156)
Phi(64,166) Phi(64,168) Phi(64,170) Phi(64,172) Phi(64,174) Phi(64,178)
Phi(64,186) Phi(64,188) Phi(64,192) Phi(64,194) Phi(64,196) Phi(64,198)

Phi(72,18) Phi(72,36) Phi(72,150)

Phi(80,16) Phi(80,48) Phi(80,80) Phi(80,112)

Phi(81,21) Phi(81,153)

Phi(96,4) Phi(96,20) Phi(96,28) Phi(96,36) Phi(96,44) Phi(96,52) Phi(96,116)
Phi(96,124) Phi(96,125) Phi(96,132) Phi(96,140) Phi(96,147) Phi(96,148)
Phi(96,164) Phi(96,172) Phi(96,176) Phi(96,188) Phi(96,192) Phi(96,200)


Phi(112,16) Phi(112,48) Phi(112,144) Phi(112,176)

Phi(121,8) Phi(121,40) Phi(121,88) Phi(121,104) Phi(121,120) Phi(121,136) 
Phi(121,152) Phi(121,168) Phi(121,184) Phi(121,200)

Phi(128,2) Phi(128,4) Phi(128,6) Phi(128,8) Phi(128,10) Phi(128,12)
Phi(128,14) Phi(128,16) Phi(128,20) Phi(128,22) Phi(128,24) Phi(128,28)
Phi(128,30) Phi(128,32) Phi(128,34) Phi(128,38) Phi(128,40) Phi(128,44)
Phi(128,46) Phi(128,48) Phi(128,54) Phi(128,56) Phi(128,64) Phi(128,66)
Phi(128,68) Phi(128,70) Phi(128,74) Phi(128,76) Phi(128,78) Phi(128,80)
Phi(128,82) Phi(128,86) Phi(128,88) Phi(128,90) Phi(128,92) Phi(128,94)
Phi(128,102) Phi(128,106) Phi(128,108) Phi(128,110) Phi(128,112) Phi(128,120)
Phi(128,126) Phi(128,128) Phi(128,130) Phi(128,132) Phi(128,134) Phi(128,136)
Phi(128,142) Phi(128,146) Phi(128,148) Phi(128,150) Phi(128,152) Phi(128,154) 
Phi(128,156) Phi(128,160) Phi(128,162) Phi(128,164) Phi(128,166) Phi(128,170)
Phi(128,172) Phi(128,174) Phi(128,176) Phi(128,180) Phi(128,182) Phi(128,184)
Phi(128,186) Phi(128,188) Phi(128,190) Phi(128,192) Phi(128,194) Phi(128,200)

Phi(144,6) Phi(144,24) Phi(144,25) Phi(144,49) Phi(144,80) Phi(144,112) 
Phi(144,144) Phi(144,176) Phi(144,200)


Given that Dubya has control of a such vast arsenal, I'm sure 
the most pressing issue on his mind is :

Which bombs would Jesus drop?     (-- "mm")

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