Problems & Puzzles: Puzzles

Puzzle 183. Perhaps a very simple question

The quantity of composite numbers that are not expressible as a sum of two or more consecutive primes, is finite or infinite?

The first ten numbers of this sequence, including 0 & 1, are: 0, 1, 4, 6, 9, 14, 16, 20, 21, 22. See EIS A50940.

Solution:

Felice Russo and Enoch Haga have counted - each by his own - the quantity of numbers belonging to this sequence up to several powers of ten:

 Up to Enoch Russo 10 5 5 10^2 47 47 10^3 520 520 10^4 5191 5190 10^5 51462 51460

Disregarding the small differences in their respective results, both believe that these preliminary results are indicative that the quantity of these numbers in this sequence is infinite.

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By my side I find very interesting the possibility that the fraction of the numbers belonging to this sequence could to be 1/2 as n goes to infinite; and if so happens why it happens?...

Then, we are needing urgently or another three entries in the table or a theoretical approach to our original question.

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At least one theoretical approach!

Faride FiroozBakht wrote (27/6/2002) a clever, nice & simple argument:

I had an argument about puzzle 183, "Perhaps a very simple question". In fact I have proved that there exists infinitely many [odd?] composites not expressible as the sum of an even number of consecutive primes, which is a special case of the problem.

For every k>1 let us define f(k) as the smallest odd multiple of 3 which is greater than

2k
SUM (p_i) , and p_i is the i'th prime number. Then it is obvious that
i=1

2k                       2k+1
SUM (p_i) < f(k) < SUM (p_i)
i=1                      i=1

and f is an increasing function. then f(k) can't been expressed as the sum of an even number of consecutive primes, because it is greater than the sum of them. And for every k, there exists a unique f(k). so there are infinitely many f(k)'s

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I (CR) would like to point out that:

1) The SUM since i=1 is obliged by the condition of the argument of summing a even quantity of consecutive primes

2) This is exactly an argument, not a proof to our question, (but precisely I asked for an argument!....:-) because there are several ( how many?) examples of f(k) values that while they are not the sum of an even quantity of consecutive primes, they are indeed a sum of an odd quantity of consecutive primes (the first example is for k=22).

But maybe now the Faride's argument may evolve into a proof adding a bit more of work...?

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Jud McCranie wrote (30/6/2002):

My figures agree with Enoch, I get (the next 3 entries for the Table above):

10^6, 518,001
10^7, 5,205,110
10^8, 52,209,54
6

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