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Re: sums of distinct primes redux
Posted:
Sep 3, 2005 2:30 PM
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"Pubkeybreaker" <Robert_silverman@raytheon.com> writes:
> Eric Thurschwell wrote:
> > This is just a response to Dr. Derek Holt's post
> > (http://groups-beta.google.com/group/sci.math/msg/474ca1e6989a55d3?hl=en&lr=&ie=UTF-8&rnum=3)
> > at the old "sums of distinct primes" topic.
>
> A follow-on challenge.
>
> Prove or Disprove:
>
> Every sufficiently large integer is the sum of *consecutive* primes.
Find an integer k that is the sum of three consecutive primes, all
greater than 2. Call the three primes p1, p2, and p3. Consider k+1.
Obviously k+1 > p1+p2+p3. Equally obviously, k+1 is greater than the
sum of any three consecutive primes where the third is less than p3.
Since only odd numbers are prime, p2 >= p1+2 and p3 >= p2+2. For the
next prime after p3, call it p4, we know that p4 >= p3+2.
Substituting the above, we see that p2+p3+p4>= p1+2 + p2+2 + p3 +2 but
this latter expression is just k+6 and k+1 < k+6. Any sum of three
consecutive primes involving primes after p4 would be even larger.
Therefore, if k is the sum of three consecutive primes then k+1 cannot
be.
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