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?
Can you argue
about this?
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.
***
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.
***
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
***
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...?
***
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, 52,209,546
***
