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Primevens and primeodds (Posted on 2016-01-10) Difficulty: 4 of 5
Let a primeven be a positive integer that is the product of an even number of primes. Let a primeodd be a positive integer that is the product of an odd number of primes. Then, 1 is a primeven because it is the product of 0 primes. 2 is a primeodd because it is the product of 1 prime. 3 is a primeodd because it is the product of 1 prime. 4 is a primeven because it is the product of 2 primes. Here are the first 10 positive integers.
Number:   Factorization:      Number of primes:   Type:
1                             0                   primeven
2         2                   1                   primeodd
3         3                   1                   primeodd
4         2*2                 2                   primeven
5         5                   1                   primeodd
6         2*3                 2                   primeven
7         7                   1                   primeodd
8         2*2*2               3                   primeodd
9         3*3                 2                   primeven
10        2*5                 2                   primeven
Suppose the primevens and primeodds had a race. First, the primevens would be ahead because 1 is a primeven. Then, there would be a tie because 2 is a primeodd. Then, the primeodds would be ahead because 3 is a primeodd. Then, there would be a tie because 4 is a primeven. Here are the winners from 1 to 10.
Number:   Type:          Primevens:     Primeodds:     Winner:
1         primeven       1              0              primevens
2         primeodd       1              1              tie
3         primeodd       1              2              primeodds
4         primeven       2              2              tie
5         primeodd       2              3              primeodds
6         primeven       3              3              tie
7         primeodd       3              4              primeodds
8         primeodd       3              5              primeodds
9         primeven       4              5              primeodds
10        primeven       5              5              tie
The primevens were ahead at the start, but have not been ahead since then. Do the primevens ever become the winner again?

No Solution Yet Submitted by Math Man    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution re(4): some stats extended | Comment 6 of 8 |
(In reply to re(3): some stats extended by Charlie)

It took about 24 hours but finally:


            evens       odds
906150257 453075129 453075128***
906150258 453075130 453075128***
906150259 453075131 453075128***
906150260 453075132 453075128***
906150261 453075132 453075129***
906150262 453075133 453075129***
906150263 453075134 453075129***
906150264 453075135 453075129***
906150265 453075135 453075130***
906150266 453075135 453075131***
...

so the answer is yes.

  Posted by Charlie on 2016-01-11 22:06:41
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