Proof that Sylvester numbers, when reduced modulo 864 , form an arithmetic progression...
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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 pro...