[seqfan] Re: Floretions, sufficient condition for conjecture

[Creighton Kenneth Dement]

Dear Seqfans,

I recently gave a list of five open conjectures.

One of those conjectures is this one:

X in Z^{infty} if and only if 4*tesseq(X) is a sequence of integers.

For those who do not wish to spend time reading through my definitions, I
would like to give one purely algebraic formulation of this conjecture- or
at least what would be a sufficient condition for it to be correct. Anyone
who can prove it one way or another would definitely be saving me some
sleep!

***********

Let A, B, C, ..., N, O, P be rational numbers with numerators which are
integers and denominators which are powers of 2. Assume that
4*A or 4*B or ... or 4*N or 4*O is not an integer.

Furthermore, assume that the following list of terms is not equal zero.

+1.0*JJ +1.0*PP -1.0*AA +1.0*LL
-1.0*FF +1.0*II +1.0*NN +1.0*KK
-1.0*EE +1.0*MM +1.0*GG +1.0*OO
-1.0*BB +1.0*HH -1.0*CC -1.0*DD
-2.0*FK -2.0*GD -2.0*EJ +2.0*PA
-2.0*MF +2.0*BP -2.0*DL -2.0*HE
+2.0*CP -2.0*FI -2.0*EO -2.0*DN
-2.0*LB +2.0*PD -2.0*CN -2.0*GA
-2.0*AJ -2.0*OC -2.0*HB +2.0*EP
+2.0*PF -2.0*KA -2.0*MB -2.0*CI
+2.0*HI -2.0*OM +2.0*DA +2.0*PG
-2.0*NK +2.0*PH +2.0*EB +2.0*IG
+2.0*FC -2.0*LJ +2.0*IP +2.0*GH
+2.0*MN +2.0*AE -2.0*LI +2.0*JP
+2.0*OL -2.0*HN +2.0*KP +2.0*AF
+2.0*KO +2.0*BD -2.0*JI +2.0*PL
+2.0*JN +2.0*BF -2.0*GO +2.0*MP
-2.0*KH +2.0*MJ +2.0*NP +2.0*DC
+2.0*KL -2.0*MG +2.0*OP +2.0*EC

Then it follows that either 4*Q or 4*R or 4*S is not an integer where

Q =

+1.0*JJ +1.0*PP -1.0*AA +1.0*LL -1.0*FF +1.0*II +1.0*NN +1.0*KK
-1.0*EE +1.0*MM +1.0*GG +1.0*OO -1.0*BB +1.0*HH -1.0*CC -1.0*DD


R =

+6.0*BFM +3.0*PHH +6.0*LBD -3.0*PFF +6.0*FCI +6.0*GAD +6.0*DCN
-3.0*EEP -3.0*PAA +6.0*JNM +6.0*KAF +3.0*JPJ +6.0*HEB +3.0*IPI -3.0*BPB
-6.0*NKH +3.0*MPM -3.0*PDD -6.0*LJI +6.0*IGH +1.0*PPP +6.0*KOL -3.0*CPC
+3.0*NPN +3.0*OOP +6.0*OCE +3.0*PLL +3.0*GGP +6.0*AEJ +3.0*KKP -6.0*MGO


S =

+6.0HHGG +6.0EBEB +2.0HHLL +2.0EEFF +2.0IIMM -8.0AJHB -6.0PFPF +1.0CCCC
+2.0JJHH -8.0HEFM +6.0IGGI +6.0PPOO -2.0AANN +24.0HEBP +1.0BBBB -8.0LIEA
+2.0CCAA -2.0LLEE +1.0OOOO -8.0HEDL +6.0NNKK -6.0DNDN +6.0AADD -2.0EEII
+8.0EBIG -8.0GABL -2.0HHFF +24.0PAGD -6.0BBMM +24.0JNMP +6.0KPPK -8.0LIMN
-8.0NKBE +2.0MMKK -2.0MMDD -24.0MPGO -2.0NNEE -8.0EJKF -6.0EEOO +6.0GOOG
+2.0MMLL -24.0LJIP +8.0OLFA -2.0FFLL +6.0PHHP -2.0CCHH -8.0DNOE +6.0PPII
-6.0MFFM +6.0OOKK -8.0DCHK -2.0BBGG +6.0PLLP -2.0IIDD -8.0FCJL +8.0MNEA
+8.0JNBF -24.0NKHP -2.0AAOO -2.0OOBB +6.0MPMP +1.0HHHH -2.0AAMM -8.0GOJN
+6.0ECCE -8.0GDFK -6.0IICC -6.0HHEE -2.0LLCC +8.0GHFC -2.0IIAA -2.0CCKK
+2.0EEDD -8.0KOJI -2.0IIBB -6.0CNCN +6.0JNJN +8.0HIDA +2.0GGJJ +2.0HHMM
-2.0NNFF +6.0DCDC +2.0AABB -8.0ECMG +1.0MMMM +6.0LILI -2.0LLAA -8.0IGKN
+1.0FFFF +6.0IIJJ +6.0MNMN -6.0KAKA -6.0FFII +24.0AEJP +2.0JJKK -2.0DDKK
-8.0MBIC +2.0NNII -6.0EJJE -2.0NNBB -8.0CNLB +24.0BFMP +6.0LJJL +24.0KAFP
-8.0AFNH +2.0OOJJ +2.0GGKK -2.0OODD +2.0GGNN +1.0AAAA -8.0BFOG -8.0KHMJ
-6.0PPBB -2.0OOFF -6.0OOCC +2.0BBCC +8.0DCMJ +6.0BFBF -2.0GGFF +2.0FFDD
+6.0AFFA -8.0KLMG -2.0FFJJ -8.0GDEJ -6.0GAGA +24.0IPCF +1.0IIII -2.0JJCC
+6.0BDBD -2.0BBKK -2.0CCGG +1.0JJJJ -6.0DLLD +6.0JPPJ +6.0HHKK -6.0AAJJ
-2.0HHAA -8.0CNAG +1.0DDDD -2.0CCMM +1.0EEEE +24.0EPOC +6.0MGMG -6.0HBBH
+24.0KOLP -8.0OCAJ +8.0ECKL +6.0AEAE +6.0PPNN -2.0BBJJ +24.0HIPG -2.0GGEE
+6.0PPGG -2.0MMEE -8.0DAOM +6.0HHII -8.0JIBD +6.0OOMM +2.0IIKK +2.0IIOO
+2.0NNOO +1.0GGGG -6.0PPEE +2.0GGLL -2.0JJDD +6.0MJJM +24.0DNPC -8.0KAMB
-6.0KKFF +8.0KOBD +24.0LBDP +6.0NNHH -6.0PAPA -6.0PDPD +2.0NNLL -8.0OLHN
-8.0CIAK +2.0OOHH +1.0LLLL -2.0KKEE +1.0KKKK -8.0DNIF +1.0NNNN +6.0FFCC
-6.0LLBB +1.0PPPP -8.0LJHG -8.0FIEO +6.0KKLL -2.0HHDD -6.0CPCP +6.0OOLL
-8.0MFLD -6.0GDDG -8.0HIMO -8.0OCBH


Many thanks and sincerely,
Creighton