Relate Mersenne Primes To Sq Triangular Nos.

[ramsey2879]

Conjecture, For p>2, the $$2^(p-1)$$ th square triangular number is divisible by $$M_{p}$$ if and only if $$M_{p}$$ is prime. I checked this for 2<p<27. For instance the first four square triangular numbers are 0,1,36 and 1225 and the fourth is divisible by .
PS In fact it appears that if is prime then for any starting integers $$S_{1}$$ and $$S_2$$ having the recursive relation ,
$$S_{n} = 6*S_{n-1} - S_{n-2}$$ the following congruence holds:
[tex]S_{2^{p-1}} = S_{1} \mod M_{p}. There is prize money lurking here for those who are interested.

_________________
Have a useful day

 

[Zurtex]

Fancy explaining that a bit more clearly, like what you mean by a square triangular number?

 

[Nimz]

Looks like he means numbers that are simultaneously square numbers and triangular numbers. E.g. 36 = 6x6 = 1+2+3+4+5+6+7+8

 

[ramsey2879]

Fancy explaining that a bit more clearly, like what you mean by a square triangular number?

The recursive relation for square triangular numbers is known to be $$S_0 = 0, S_1 = 1, S_{n} = 6*S_{n-1} - S_{n-2}$$
$$T(m) = m(m+1)/2 = S_{n}*S_{n}$$

Thus my conjecture about square triangular numbers is just a special case of my more general conjecture which is that for the same type of series with any two starting integers that if $$M_{p}$$ is a prime Mersenne number then $$S_{(1+i)} = S_{(2^{p-1}+i)} \mod M_{p}$$. With $$S_{1} = 0, M_{p} = \mbox{prime}$$ then $$M_{p} \mid S_{2^{p-1}}$$ for any integer S_2.

 

[ramsey2879]

Fancy explaining that a bit more clearly, like what you mean by a square triangular number?

The recursive relation for square triangular numbers is known to be $$S_0 = 0, S_1 = 1, S_{n} = 6*S_{n-1} - S_{n-2}$$
$$T(m) = m(m+1)/2 = S_{n}*S_{n}$$

Thus my conjecture about square triangular numbers is just a special case of my more gemeral conjecture which is that for the same type of series with any two starting integers that if $$M_{p}=2^{p} - 1$$ is a prime Mersenne number then $$S_{(1+i)} = S_{(2^{p-1}+i)} \mod M_{p}$$. With $$S_{1} = 0, M_{p} = \mbox{prime}$$ then $$M_{p} \mid S_{2^{p-1}}$$ for any integer S_2.

 

[ramsey2879]

The recursive relation for square triangular numbers is known to be $$S_0 = 0, S_1 = 1, S_{n} = 6*S_{n-1} - S_{n-2}$$
$$T(m) = m(m+1)/2 = S_{n}*S_{n}$$

Thus my conjecture about square triangular numbers is just a special case of my more gemeral conjecture which is that for the same type of series with any two starting integers that if $$M_{p}=2^{p} - 1$$ is a prime Mersenne number then $$S_{(1+i)} = S_{(2^{p-1}+i)} \mod M_{p}$$. With $$S_{1} = 0, M_{p} = \mbox{prime}$$ then $$M_{p} \mid S_{2^{p-1}}$$ for any integer S_2.

After more consideration, I realized that it is not necessary to store more than one term and that my test can be refined so that there are only p-1 terms that need to be calculated.

Input p
M(p) = 2^p -1
S_0 = 1
C = p
Do
S_0 = (4*S_0)*(8*S_0 + 1) Mod M(p)
C = C - 1
Loop until C = 1
If S_0 = 1 Then
Print "M(" & p & ") is prime"
Else
Print "M(" & p & ") is not prime"
End If.

Although this test involves more steps, it basically involves almost the same complexity as the Lucus-Lehmer Test since the additional steps are register shifts and the addition of 1
It is based upon the fact that 1 is the 2nd square triangular number and that The S_0's follow the sequence ST(2),ST(3),ST(5),ST(9),ST(17)... i.e. ST(2^n+1) in the sequence of square triangular numbers (modulus M(p) of course). I checked the test for p = 3 to 28. It doesn't work for p = 2 since 3 is a factor of 6.

Any comments?

 

[ramsey2879]

I managed to write a basic program using true/false arrays with binary like addition subtraction and multiplication that checked all Mersenne numbers, $$2^{p} -1 \mid p = \mb{prime}$$ for primes from p = 3 up through p = 929 base upon the relation between Mersenne Primes and Square Triangular Numbers. I designed it to do primes up to p= 4999 without modification, but it takes something in the order of a 8 hour day to check up through p=929. I am not sure since I upgraded my program after the first 82 primes and it did the next 82 primes in about the same time, so I am only estimating that it would have taken less than 1/4 the time to check the first 82 primes.

My conjecture in the preceeding post still holds with the lone exception being 2^2 - 1 which I don't think affects my conjecture since 3 is a special case related to the recursive sequence S_n = 6*S_(n-1) - S_(n-2) for square triangular numbers.

It probably would be fruitless to explore my conjecture further in this mannor. A better approach is to find a proof. Any takers or suggestions.