20050913, 07:00  #1 
Jun 2003
3^{2}×5^{2}×7 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 
20050913, 07:24  #2 
Jun 2003
3^{2}×5^{2}×7 Posts 
There are multiple solution till each p but the smallest t(1) will be the correct one.
Citrix 
20050914, 13:07  #3 
Sep 2005
UGent
2^{2}×3×5 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 
20050914, 23:00  #4 
Jun 2003
3^{2}×5^{2}×7 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 
20050914, 23:23  #5 
Jun 2003
3^{2}·5^{2}·7 Posts 
Also prime for 2^(2^6)

20050914, 23:33  #6 
Jun 2003
3^{2}×5^{2}×7 Posts 
no primes upto n=18

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