# Triangle primes

Max maxale at gmail.com
Sat Apr 15 02:49:42 CEST 2006

```Some other terms are also missing, e.g.:

389 = 16*17/2 + 22*23/2.

A complete list of primes up to 1000 being the sum of two triangular
numbers is the following:

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137,
139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281,
307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487,
499, 541, 571, 577, 587, 601, 619, 631, 659, 661, 673, 709, 727, 739,
751, 757, 769, 821, 823, 839, 853, 877, 883, 911, 919, 947, 967, 991,
997

Max

On 4/14/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
>
> Do you have something against 1?
>
> 2 = 1+1, 7 = 6+1, 11=10+1, 29=28+1, 37=36+1, ....
>
>
>
> -----Original Message-----
> From: Andrew Plewe <aplewe at sbcglobal.net>
> To: 'seqfan' <seqfan at ext.jussieu.fr>
> Sent: Fri, 14 Apr 2006 17:30:55 -0700
> Subject: Triangle primes
>
>
> I plan to submit the following sequence to the OEIS. These are Triangle
> primes, or primes which can be expressed as the sum of two triangle numbers:
>
>
> 13, 31, 43, 61, 73, 83, 97, 101, 127, 139, 151, 157, 163, 181, 191, 193,
> 199, 211, 227, 241, 263, 281, 307, 331, 353, 367, 373, 379, 409, 421, 433,
> 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661, 673, 709,
> 727, 739, 751, 757, 769, 821, 823, 839, 853, 877, 883, 911, 919, 967, 991,
> 997, 1033, 1039, 1051
>
>
> Would someone be so kind as to verify that the terms I have are correct? I
> have checked these by hand by referencing a list of the first 1000 primes
> from the Prime Pages website, a process which is prone to errors.
>
>
> I find this sequence interesting for several reasons. First, more primes
> than I'd originally expected show up in the list. Second, I derived this
> list from a table of sums of triangle numbers. That table seems to have some
> interesting properties. One of these is a simple prime finding/proving
> method which may or may not (I don't have a proof yet) work for all odd
> integers in the table. The method is essentially the following:
>
>
> 1.) Look for a Sophie Germain "pair" (i.e. n and 2n +/- 1). The product of
> any Sophie Germain "pair" is also a triangle number. For my example I'll use
> 3779 * 1889
>
> 2.) Find another number which is prime. In this case I'll use 5197. Divide
> that number plus or minus one by two, the dividend should be an even number.
> 5196 / 2 = 2598 will work. The product of this "pair" is also a triangle
> number.
>
> 3.) Add the two triangle numbers. 3779 * 1889 + 5197 * 2598 = 20640337. This
> is our prime "candidate".
>
> 4.) Perform a GCD with the candidate and the two triangle numbers. If GCD >
> 1, the number is composite. In this case, GCD = 1 for both triangle numbers.
>
> 5.) Find all of the "neighboring" triangle sums immediate around the
> candidate value. This can be done by finding the triangle numbers immediate
> before and after our "pairs",:
>
>
>  3777 * 1889
>  3779 * 1889
>  3779 * 1890 and
>
>  5195 * 2598
>  5197 * 2598
>  5197 * 2599
>
> and adding together all possible combinations of the pairs (excluding adding
> a pair to itself):
>
> 3777 * 1889 + 5195 * 2598 = 20631363
> 3777 * 1889 + 5195 * 2599 = 20635141 ... etc.
>
> 6.) Now, find the difference of each neighbor and the candidate value:
>
>  20640337 - 20631363 = 8974
>  20640337 - 20635141 = 5196 ... etc.
>
> 7.) Create a set of sums and differences of these values:
>
>  8974 - 5196 = 3778
>  8974 + 5196 = 14170 ... etc.
>
> 8.) GCD all values in the set of sums and differences with the candidate
> value. If a gcd > 1 is found, the number is composite. if not, the number is
> prime. In this case, the number is composite:
>
>  8976 + 5196 = 14172, gcd(14172, 20640337) = 1181
>
>
> Another interesting thing (to me, anyway) is that if there are not an
> infinite number of Sophie Germain primes then at some point in time the
> "density" of primes in the sum table of triangle numbers will drop
> drastically. Anyway, I apologize for the length of this email if you're
> bored to tears by this, hopefully some of you find this interesting. Thanks!
>
>  -Andrew Plewe-
>
>
>
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