The following observation has been made:
Numbers in Sylvester's sequence,when reduced $modulo 864$, form an arithmetic progression, namely $$7,43,79,115,151,187,223,259,295,331,.....$$
This has been checked for the first ten members of the sequence:
$$7≡7(mod864)$$ $$43≡43(mod864)$$ $$1807≡79(mod864)$$ $$3263443≡115(mod864)$$ $$10650056950807≡151(mod864)$$ $$113423713055421844361000443≡187(mod864)$$ $$12864938683278671740537145998360961546653259485195807≡223(mod864)$$
I have been unable to check other numbers in this sequence, due to the rapid growth of the sequence, the numbers become too large to handle. However, we can use congruence relations, congruence arithmetic and arithmetic of residue classes to prove that Sylvester numbers ,when reduced $modulo 864$, form an arithmetic progression. Consider the following: One may define the sequence by the recurrence relation:
$$si=si−1(si−1−1)+1$$
Sylvester's sequence can also be defined by the formula:
$$sn=1+∏n−1i=0si$$
$$7≡7 (mod864)$$ $$7x6+1=43≡43 (mod864)$$ $$43x42+1=1807≡79 (mod 864)$$ $$79x78+1=6163≡115 (mod 864)$$ $$115x114+1=13111≡151 (mod 864)$$ $$151x150+1=22651≡187 (mod 864)$$ $$187x186+1=34783≡223 (mod 864)$$ $$223x222+1=49507≡259 (mod 864)$$ $$259x258+1=66823≡295 (mod 864)$$ $$295x294+1=86731≡331 (mod 864)$$ $$331x330+1=109231≡367 (mod 864)$$ $$367x366+1=134323≡403 (mod 864)$$ $$403x402+1=162007≡439 (mod 864)$$ $$439x438+1=192283≡475 (mod 864)$$ $$475x474+1=225151≡511 (mod 864)$$ $$511x510+1=260611≡547 (mod 864)$$ $$547x546+1=298663≡583 (mod 864)$$ $$583x582+1=339307≡619 (mod 864)$$ $$619x618+1=382543≡655 (mod 864)$$ $$655x654+1=428371≡691 (mod 864)$$ $$691x690+1=476791≡727 (mod 864)$$ $$727x726+1=527803≡763 (mod 864)$$ $$763x762+1=581407≡799 (mod 864)$$ $$799x798+1=637603≡835 (mod 864)$$ $$835x834+1=696391≡7 (mod 864)$$ $$7x6+1 =43≡43 (mod 864)$$ $$43x42+1 =1807≡79 (mod 864)$$ $$79x78+1=6163≡115 (mod 864)$$ etc.
Notice that after $24$ cycles , we get back to where we started. Hence Sylvester numbers , reduced $modulo 864$ form an arithmetic progression of $24$ terms which will then repeat until infinity. Therefore Sylvester sequence , reduced $mod 864$, forms an arithmetic progression of $24$ terms, which will repeat until infinity. QED