Mini-challenge: First Prime of form 403333...

[Valentin Albillo]

 
Hi, Thomas:

(11-14-2016 09:02 AM)Thomas Ritschel Wrote:  Thanks for the detailed analysis!

You're welcome, but I wouldn't call my post an analysis, more like a few observations I made on the go while solving the challenge.

Quote:[...]every second number of the sequence 403333... (those with an even number of digits) has algebraic factors:

           4033 = 109 * 37
        403333 = 1099 * 367 = (7*157) * (367)
        40333333 = 10999 * 3667 = (17*647) * (19*193)
        4033333333 = 109999 * 36667 = (317*347) * (37*991)

There is obviously a pattern in the factors.

Interesting. Did you notice this pattern yourself or did you read about it someplace ? In that case, I would appreciate a link to it.

Quote:Thus, in general, one can skip the even exponents; in your case the odd indices. Together with the "mod 6" congruences this reduces to testing only the case index = 4 mod 6.

Great. This way, instead of having to check 483 candidates, only about 60-70 will need to be checked out, an 800% speed up.

Thanks a lot for posting those really interesting facts, a real textbook example of how a little analysis can do much better than relying in just pure brute force.

Regards.
V.