theory on Mersenne primes ?

[science_man_88]

Originally Posted by science_man_88

To expand it going to 5 we get 5-1 = 4 = 1 +3 which means since both 1 and 3 are triangular 5 is on a diagonal with 1 if you break it to the next corner around a number spiral after the original number you get the amount of squares: 1,1,2,2,3,3,4,4,5,5 . This is why the formula can tell you were it is because mod the sum of triangular numbers. tells you where in relation to a specific corner the number is and,the fact that given enough numbers you can cover any square eventually. 3 and 13 are both at the top right corner in their specific parts of the spiral so they are on the same diagonal.

since 3 corners are needed we can tell that if 1 is in the center of a square 5 is in the top left corner, going counter clockwise.. 2 corners are needed for 3 so it would be top right corner. from this we can deduce 3 and 5 must both be in corners 2 and 3 which are in a straight line from each other so 3 and 5 are in a straight line.

[CRGreathouse]

Originally Posted by science_man_88

3-1 = 2 = 1 + 1 ( largest sum of 2 triangular numbers that is less than or equal to P-1) since it would be congruent to 0 mod (the sum of those triangular numbers) we can take it that 3 is on the same diagonal as 1.Knowing something like this could help pinpoint where 2 numbers are in relation to each other ( proving if they are in a diagonal together or not). Even negative moduli could be useful for certain diagonals and if they are equal to .5* next integer to use in for 2 numbers assuming they are on the same or directly opposing sides they are likely in a line with each other.

Originally Posted by science_man_88

To expand it going to 5 we get 5-1 = 4 = 1 +3 which means since both 1 and 3 are triangular 5 is on a diagonal with 1 if you break it to the next corner around a number spiral after the original number you get the amount of squares: 1,1,2,2,3,3,4,4,5,5 . This is why the formula can tell you were it is because mod the sum of triangular numbers. tells you where in relation to a specific corner the number is and,the fact that given enough numbers you can cover any square eventually. 3 and 13 are both at the top right corner in their specific parts of the spiral so they are on the same diagonal.

Originally Posted by science_man_88

since 3 corners are needed we can tell that if 1 is in the center of a square 5 is in the top left corner, going counter clockwise.. 2 corners are needed for 3 so it would be top right corner. from this we can deduce 3 and 5 must both be in corners 2 and 3 which are in a straight line from each other so 3 and 5 are in a straight line.

I don't know what you're getting at. You're expressing a number as n = a + b + c with a and b triangular and c > 0 minimal; ok. (Here's code:)

prectri(n)={
	my(m=sqrtint(n+=n));
	if(m*(m+1)<=n,m*(m+1),m*(m-1))/2
};
addhelp(prectri, "prectri(n): Largest triangular number less than or equal to n.");

prectriM(n)={
	my(m=sqrtint(n+=n));
	if(m*(m+1)<=n,m,m-1)
};
addhelp(prectriM, "prectriM(n): Index of largest triangular number less than or equal to n.");

twoTri(n)={
	my(b=n,bAt,t);
	for(m=0,prectriM(n\2),
		t=n-m*(m+1)/2;
		t-=prectri(t);
		if(t<b,b=t;bAt=m)
	);
	[bAt*(bAt+1)/2,prectri(n-bAt*(bAt+1)/2)]
};
addhelp(twoTri, "twoTri(n): Gives two triangular numbers the sum of which is the largest sum of two triangular numbers <= n.");

sm88Partition(P)={
	my(t=twoTri(P-1));
	concat(t,P-t[1]-t[2])
};

Then you say, "since it would be congruent to 0 mod (the sum of those triangular numbers)", but what is "it"? In your example the sum of the triangular numbers is 2 so the "it" is even, so we know the "it" isn't your example number 3. But what is it, then?

Then you say that this could "pinpoint" something, or be "useful for certain diagonals", but what would be pinpointed and what does "useful for certain diagonals" even mean?

Then you say, "the formula can tell you were it is" (I assume you mean "where"), but what is this "it"?

And generally, what does all this analysis give us?

[science_man_88]

Originally Posted by CRGreathouse

I don't know what you're getting at. You're expressing a number as n = a + b + c with a and b triangular and c > 0 minimal; ok. (Here's code:)

prectri(n)={
	my(m=sqrtint(n+=n));
	if(m*(m+1)<=n,m*(m+1),m*(m-1))/2
};
addhelp(prectri, "prectri(n): Largest triangular number less than or equal to n.");

prectriM(n)={
	my(m=sqrtint(n+=n));
	if(m*(m+1)<=n,m,m-1)
};
addhelp(prectriM, "prectriM(n): Index of largest triangular number less than or equal to n.");

twoTri(n)={
	my(b=n,bAt,t);
	for(m=0,prectriM(n\2),
		t=n-m*(m+1)/2;
		t-=prectri(t);
		if(t<b,b=t;bAt=m)
	);
	[bAt*(bAt+1)/2,prectri(n-bAt*(bAt+1)/2)]
};
addhelp(twoTri, "twoTri(n): Gives two triangular numbers the sum of which is the largest sum of two triangular numbers <= n.");

sm88Partition(P)={
	my(t=twoTri(P-1));
	concat(t,P-t[1]-t[2])
};

Then you say, "since it would be congruent to 0 mod (the sum of those triangular numbers)", but what is "it"? In your example the sum of the triangular numbers is 2 so the "it" is even, so we know the "it" isn't your example number 3. But what is it, then?

Then you say that this could "pinpoint" something, or be "useful for certain diagonals", but what would be pinpointed and what does "useful for certain diagonals" even mean?

Then you say, "the formula can tell you were it is" (I assume you mean "where"), but what is this "it"?

And generally, what does all this analysis give us?

well the formula gives a way to prove numbers line up without making the number spiral. okay technically it's 1 mod the sum to get to a corner. 3 is in a corner 1+1 is a sum of triangular numbers +1 ( to count the center( not counted in formula)) and you get 3 which means 3 is in a corner. if we can prove that combinations of 2 mersenne prime exponents line up with the formula then we might be able to use knowledge of number spirals to further the search for Mersenne prime exponents.

[science_man_88]

an example of using the formula is:

20996011-1 = 447 Mod (10495071 + 10499653) = 447 Mod (A000217(4581)+A000217(4582))

so because these aren't the same n values into A000217 we know that it appears the second time 4582 appears as a length which means it's on the left side of the center. Since 447 is less than .5* 10499653 we know it's in the top left quadrant if you will. We can try and find a line to another exponent already known if none is found maybe it links to a higher exponents (yes I know maybe is not for sure).

[science_man_88]

13466917 is upper right 1686 into the 3670 I may have messed up my earlier example for the other one.

[science_man_88]

6972593 is lower left. 3021377 upper left. want more specifics ?

[science_man_88]

(19:50)>for(i=1,#mersenne1,for(y=1,#triangle,if(((mersenne1[i]-1)/2)<=triangle[y],print(mersenne1[i]","y);break())))
2,1
3,1
5,2
7,2
13,3
17,4
19,4
31,5
61,8
89,9
107,10
127,11
521,23
607,25
1279,36
2203,47
2281,48
3217,57
4253,65
4423,66
9689,98
9941,100
11213,106
19937,141
21701,147
23209,152
44497,211
86243,294
110503,332
132049,363
216091,465
756839,870
859433,927
1257787,1122
1398269,1182
2976221,1725
3021377,1738
6972593,2641
13466917,3670
20996011,4582

my latest help on figuring where they are so far and if they line up.

I forgot to highlight 89 in my file doh lol. but unless someone feels safe to download it it's unimportant.

[CRGreathouse]

Originally Posted by science_man_88

want more specifics ?

Yes. You've spent a lot of words and many posts to discuss this method for determining whether two numbers are on the same diagonal or not. (I don't quite understand it, but I don't blame you, at least not entirely -- it's partially due to my lack of interest.) But what do we do with that, once we know it?

That is, suppose we had a gp function onTheSameDiagonal(n1, n2). (Actually, I think you should write this, possibly using the functions I wrote for you above.) How does this help us find suitable exponents?

Or, if it has a different purpose, what?

[science_man_88]

Originally Posted by CRGreathouse

Yes. You've spent a lot of words and many posts to discuss this method for determining whether two numbers are on the same diagonal or not. (I don't quite understand it, but I don't blame you, at least not entirely -- it's partially due to my lack of interest.) But what do we do with that, once we know it?

That is, suppose we had a gp function onTheSameDiagonal(n1, n2). (Actually, I think you should write this, possibly using the functions I wrote for you above.) How does this help us find suitable exponents?

well we know it has to be prime if they have to either fit same diagonal or column / row in a table then we can use one to find another though not necessarily in order.

okay:

(20:56)>b=3;for(i=2,100,b=b+2*i;if(isprime(b),print(b)))
7
13
31
43
73
157
211
241
307
421
463
601
757
1123
1483
1723
2551
2971
3307
3541
3907
4423
4831
5113
5701
6007
6163
6481
8011
8191
9901
(20:56)>b=5;for(i=2,100,b=b+2*i+1;if(isprime(b),print(b)))
17
37
101
197
257
401
577
677
1297
1601
2917
3137
4357
5477
7057
8101
8837

these check for all primes on diagonals, from the corners. your twotri looks good for figuring out where they are in the spiral then we have to check on the diagonals( 1 by 1) and straight lines, to see if they connect. if some don't with the exponents known so far then my idea suggest there are further exponents that these would connect to.

[CRGreathouse]

Originally Posted by science_man_88

well we know it has to be prime if they have to either fit same diagonal or column / row in a table then we can use one to find another though not necessarily in order.

You're printing A002383, the primes in A002061. What does this have to do with, uh, anything?

[science_man_88]

Originally Posted by CRGreathouse

You're printing A002383, the primes in A002061. What does this have to do with, uh, anything?

these are the primes on the diagonals that 3 and 5 reside on if 1 is at the center.