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Binary Primes (Posted on 2004-02-18) Difficulty: 4 of 5
How many primes, written in usual base 10, have digits that are alternating 1s and 0s, beginning and ending with one?

For example (not necessarily prime):
1, 101, 10101, ...

See The Solution Submitted by Aaron    
Rating: 3.5000 (2 votes)

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Solution solution | Comment 12 of 13 |
All numbers in base-10 of alternating 1s and 0s ending in 1 are in the form (102n-1)/99.
As as (102n-1)/99 equals (10n-1)(10n+1)/99; and, as a prime is a positive integer that can have no other factors besides 1 and itself; either (10n-1) or (10n+1) must equal one of the divisors of 99 (99, 33, 11, 9, 3, 1) with the other having a prime factor and no other factors but that that composes the respective complementary divisor (1, 3, 9, 11, 33, 99).
 
One can consequently find that (10n-1) must equal 99 and (10n+1) must equal 101. Thus, the only possible number is 101, and, therefore the answer to the puzzle is there is only 1 prime in base-10 of alternating 1s and 0s ending in 1.
  Posted by Dej Mar on 2011-06-22 05:56:06
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