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 2005-09-13, 07:00 #1 Citrix     Jun 2003 30648 Posts Sequence Generate a sequence where the N+1 term is the product of all previous terms +1 and the t(N+1) is divisible byp(N+1) ie the Nth prime. SO t(1) divisible by 1 t(2)=t(1)+1 and divisble by 2 t(3)=t(2)*t(1)+1 and divisble by 3 t(4)=t(3)*t(2)*t(1)+1 and divisble by 5. and so on... Good luck this is a hard problem, lets see who can generate the longest sequence. Can you find a algorithm to generate this sequence to some prime p? Citrix
 2005-09-13, 07:24 #2 Citrix     Jun 2003 110001101002 Posts There are multiple solution till each p but the smallest t(1) will be the correct one. Citrix
 2005-09-14, 13:07 #3 Jushi     Sep 2005 UGent 3C16 Posts Let T = t(1). Then: t(2) = T + 1 t(3) = t(2)*t(1) + 1 = (T + 1)*T + 1 = T^2 + T + 1 t(4) = t(3)*t(2)*t(1) + 1 = (T^2 + T + 1)*(T + 1)*T + 1 = T^4 + 2T^3 + 2T^2 + T + 1 By looking modulo 5, the condition that t(4) is divisible by 5 is impossible. The best sequence is thus: t(1) = 1 t(2) = 2 t(3) = 3
 2005-09-14, 23:00 #4 Citrix     Jun 2003 158810 Posts cool solution, but the formula you generated for t4 is prime for 2,4,16,256. All 2^2^n. Pretty cool. Are there any other primes of the form 2^2^n? Citrix
 2005-09-14, 23:23 #5 Citrix     Jun 2003 22·397 Posts Also prime for 2^(2^6)
 2005-09-14, 23:33 #6 Citrix     Jun 2003 63416 Posts no primes upto n=18

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