Pick a positive integer to start a sequence. Now double it, and add one to the result: this is the second number of your sequence. Double that number, and add one, and that will be your third number; repeat the doubling and adding, and you will have a fourth number, and so on.

If you start with a prime number, and you keep doubling and adding one, is it possible to produce a sequence with only prime numbers?

(In reply to A bet by Federico Kereki)

This conjecture is true for all primes through 8647. The verification program crashes with an overflow on the 8659th iteration of the doubling and adding for the next prime, 8663.

list
   10   N=2
   20   while Ct<48
   25   Lp=N
   30   N=nxtprm(N)
   35   Nu=N
   40   for I=1 to N-1
   50    Nu=2*Nu+1
   60   next
   70   if Nu @ N>0 then print N,Nu:Ct=Ct+1
   90   wend
OK
run
Overflow in 50
?lp
 8647
OK
?n
 8663
OK
?i
 8659
OK

Line 70 would have shown any exception to the rule.

Posted by Charlie on 2006-11-30 09:07:59