Re: More on triangle numbers and primes!

> If this is true, there could be an easy way to check for primality for
> large integers.
> First all odd number checks will have the first two
> terms in common whether prime or composite, so
> really you could drop the first two checks.
> The first is obvious, it is just
> (n*(n+1))/2 - (n-1*(n))/2 where (n) is the first result.
> The second is also obvious, it is just n­_c = ceiling
> (n/2) then (n_c*(n_c+1))/2 - (n_c-2*(n_c-1))/2 =
> original n.
> So you only have to search the remaining half to see if
> any resulting (n) is present.
> If there is, then n is composite,if not,n is prime.
> Dan

I'm still looking at this...

Okay, n_c = ceiling (n/2) . For a convenient algebra (or perhaps for a less
convenient algebra) and with odd integer n the ceiling of (n/2) is ((n + 1)
/ 2)...


First case:


n(n + 1) - m(m + 1)
------------------- =? n
2


When m = (n - 1)


n(n + 1) - (n - 1)(n)
--------------------- = n
2



n(n + 1 - n + 1)
---------------- = n
2


and


n(2)
----
2 = n as true and yet true for all n.

___________________________________________

Second case:


a(a + 1) - b(b + 1)
------------------- =? n
2



When a = ((n + 1)/ 2) and b = (((n + 1)/ 2) - 2)



((n + 1)/ 2)*((n + 1)/ 2) + 1)-((n + 1)/ 2) - 2)*(((n + 1)/ 2) - 1)
---------------------------------------------------------
2



((n + 1)/ 2)^2 + ((n + 1)/ 2) - ((n + 1)/ 2)^2 + 3((n + 1)/ 2) - 2
----------------------------------------------------------
2



4((n + 1)/ 2) - 2
----------------- = n
2




2(n + 1) - 2 2n
------------ -- = n
2 = 2


And yeah that is really just an algebra exercise...
_____________________________________


Well, I have the proofs yet to study and also the graphical characteristics
of the triangle...

The questions are:

Is is proven that there are no solutions between this first case and the
second case ? (The original poster says calls the second case the second
solution and says that it is obvious.)

Is there anything in the logic of the proofs that allow a calculation as to
where a third solution will be (within a range) ?


.

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