## The curious number 9839389.

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• 9839389 = 7 * 43 * 97 * 337 Sum the primes between the smallest and largest prime factors: 7+11+13+...+331+337 = 10181, a prime. Sum the composites between the
Message 1 of 3 , Nov 3, 2002
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9839389 = 7 * 43 * 97 * 337

Sum the primes between the smallest
and largest prime factors:
7+11+13+...+331+337 = 10181, a prime.

Sum the composites between the
smallest and largest prime factors:
8+9+10+...+335+336 = 46751, a prime.

Can anyone find larger palindromes
like 9839389?

"In mathematics you don't understand things.
You just get used to them."
--Johann von Neumann
• ... Yes. They don t seem to be rare. The smallest is 15251=101*151 The sum of the primes 101+103+107+109+113+127+131+137+139+149+151=1367, a prime the sum of
Message 2 of 3 , Nov 3, 2002
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--- In primenumbers@y..., "Max B" <zen_ghost_floating@h...> wrote:

> Sum the primes between the smallest
> and largest prime factors:
> 7+11+13+...+331+337 = 10181, a prime.
>
> Sum the composites between the
> smallest and largest prime factors:
> 8+9+10+...+335+336 = 46751, a prime.
>
> Can anyone find larger palindromes
> like 9839389?

Yes. They don't seem to be rare.
The smallest is 15251=101*151
The sum of the primes
101+103+107+109+113+127+131+137+139+149+151=1367, a prime
the sum of the composites
(151^2-101^2+151+101)/2-1367=5059 a prime
There are more than 300 such palindromes below 2^32

The largest I can conveniently find is
4273223724=2^2*3^2*7*11*467*3301
where the sum of the primes up to and including 3301 is 700897 and
the sum of the composites 4749053 are both primes.

For large solutions to this problem I'd suggest sticking to even
palindromes.
You can divide the problem into two parts. Firstly find pairs of
primes such that the sums of the primes and composites between them
are both primes. Secondly, using the larger prime of the pair and
quite a lot of thought you should be able to solve a degree 1 modular
equation in quite a few variables to generate some candicates you can
then check for smoothness. This could be tricky. One neat way of
generating smooth palindromes is to look for palindromes with low
digit values which are divisible by other, shorter low digit value
palindromes. You can then add them together (with appropriate shifts)
to generate more palindromes of which a large part of the
factorisation is already known. For instance some of the common small
largest factors of palindromes are 101, 9091 (divides 10^5+1), 9901
(divides 10^6+1), 333667 (divides 1001001). Also 3637 but I don't
know why.
If you can find a large factor p of two low weight palindromes of
different sizes such that the sum of the primes and composites from 2
to p are prime then you're probably set for some really big solutions
to this problem.

Richard
• Using some of Richard s lucid tips below, and my usual brute-force blatant bricoleur approach (too clumsy to divulge here), I managed to find: 819870212078918
Message 3 of 3 , Nov 4, 2002
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Using some of Richard's lucid tips below, and my
usual brute-force blatant bricoleur approach
(too clumsy to divulge here), I managed to find:

819870212078918 =
2 * 7 * 11 * 13 * 17 * 17 * 10771 * 131561

where the sum of primes from 2 to 131561 is
766715749 and the sum of composites is
7887498391, both primes.

But I'm sure this solution could be easily eclipsed
by the 'heavyweights' on this list.

Thanks Richard!

----- Original Message -----
From: richard_heylen
Sent: Sunday, November 03, 2002 9:02 PM
Subject: [PrimeNumbers] Re: The curious number 9839389.

[...]

The largest I can conveniently find is
4273223724=2^2*3^2*7*11*467*3301
where the sum of the primes up to and including 3301 is 700897 and
the sum of the composites 4749053 are both primes.

For large solutions to this problem I'd suggest sticking to even
palindromes.
You can divide the problem into two parts. Firstly find pairs of
primes such that the sums of the primes and composites between them
are both primes. Secondly, using the larger prime of the pair and
quite a lot of thought you should be able to solve a degree 1 modular
equation in quite a few variables to generate some candicates you can
then check for smoothness. This could be tricky. One neat way of
generating smooth palindromes is to look for palindromes with low
digit values which are divisible by other, shorter low digit value
palindromes. You can then add them together (with appropriate shifts)
to generate more palindromes of which a large part of the
factorisation is already known. For instance some of the common small
largest factors of palindromes are 101, 9091 (divides 10^5+1), 9901
(divides 10^6+1), 333667 (divides 1001001). Also 3637 but I don't
know why.
If you can find a large factor p of two low weight palindromes of
different sizes such that the sum of the primes and composites from 2
to p are prime then you're probably set for some really big solutions
to this problem.

Richard
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