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More Consecutive Semiprimes (Posted on 2009-09-08) Difficulty: 3 of 5
Consider the series of semiprimes. Now consider the series of numbers representing the Sum Of Digits (SOD) of each semiprime.

(1.) How many times does the number '9' occur, and why?

(2.) What is the smallest semiprime that has the same SOD as the next consecutive semiprime? or prove that there is no pair of consecutive semiprimes that have the same SOD.

No Solution Yet Submitted by Larry    
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part 1 with proof | Comment 2 of 4 |

ok, to start with I'm assuming this is asking how many times does the number 9 occur as the SOD, not how many times does 9 occur as a digit in an SOD.

Lets assume there is a semiprime S>9 such that SOD(S)=9, then S is divisible by 9 and thus divisible by 3^2, but since S>9 there must be another prime divisor of S, thus S has at least 3 prime divisors.  Thus contradicting S being semiprime.  Thus the only semiprime S with SOD(S)=9 is S=9

  Posted by Daniel on 2009-09-08 12:50:03
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