Register to reply

Conjecture: Sophie Germain Triangles & x | 2y^2 + 2y - 3 = z^2

by Raphie
Tags: 16x + 9, sophie germain, triangles
Share this thread:
dodo
#19
Mar22-11, 11:28 AM
P: 688
Quote Quote by Raphie View Post
In principle, there should be one formula that generates (right to left) diagonal-specific Recursive rules for each entry in the above triangle.
If I'm reading correctly (and I'm not sure I am), the recurrence formula for your triangle entries should be
entry(n,k) = entry(n-1,k-1) + 2*entry(n-1,k)
The reason why they add up to 3^n is because that is the binomial expansion of
[itex](2+1)^n = \sum_{k=0}^n \binom n k 2^k[/itex]

Hope this helps, if this coincides with what you're going after.
Raphie
#20
Mar22-11, 02:04 PM
P: 153
Quote Quote by Dodo View Post
If I'm reading correctly (and I'm not sure I am), the recurrence formula for your triangle entries should be
entry(n,k) = entry(n-1,k-1) + 2*entry(n-1,k)
The reason why they add up to 3^n is because that is the binomial expansion of
[itex](2+1)^n = \sum_{k=0}^n \binom n k 2^k[/itex]

Hope this helps, if this coincides with what you're going after.
I am well aware of the relationship you posted, Dodo, but thank you. One can actually generate triangles very easily wherein rows sum to any power desired. e.g. A 1, 3, 1 triangle with addition rule 1A + 3B sums to powers of 4. A 1, 4, 1 triangle with addition rule 1A + 4B sums to powers of 5. etc..

What is actually being addressed on this thread at the moment are the following recursion-based relationships uncovered by Ramsey2879.

Quote Quote by ramsey2879 View Post
I can't answer your question yet but my research has shown a definite link with the Pell series

every 4th term of a pell series forms a sequence S(n) = 34*S(n-1) - S(n-2)
every 8th term of a pell series forms a sequence S(n) = 1154*S(n-1) -S(n-2)

The Sophie Germain triangular numbers were already described

A similar series of triangular numbers is

x' = {0,1,171,1326,197506,1530375,227921925,1766051596,...} having the recursive formula S(n) = 1154*S(n-2) - S(n-4) + 172

144x'+25 is a square just as 16x +9 is a square for the Sophie Germain triangular numbers.

Still More
x" = {0,1,120,1275,138601,1471470,159945555,169807226...} having the recursive formula S(n) = 1154*S(n-2) - S(n-4) + 121

576x"+49 is a square just as 16x +9 is a square for the Sophie Germain triangular numbers.

Interesting how triangular numbers relate to the Pythagorean triangle numbers and Pell series.

Regards, Kenneth Ramsey
I was merely pointing out that the terms involved in his progressions can be derived from the Pell Triangle...

16x + 9 = (4^2)x + 2*4 + 1
144x + 25 = (12^2)x + 2*12 + 1
576x + 49 = (24^2)x + 2*24 + 1
1600x + 81 = (40^2)x + 2*40 + 1
etc.

... and that it ought to be possible to generalize his results to all of the Pell Triangle in some manner as yet TBD.

For instance, Ramsey2879 is working with squares and triangles at the moment. If one were to "step over" one diagonal to the left, might there not perhaps be a way to come up with similar recursion-based formulas for multiples of tetrahedrals (perhaps in polynomial expanded form...) plus some constant that add up to cubes, pentagonal numbers or some other class of square number?

Best,
Raphie
ramsey2879
#21
Mar22-11, 09:03 PM
P: 894
Quote Quote by Raphie View Post
I am well aware of the relationship you posted, Dodo, but thank you. One can actually generate triangles very easily wherein rows sum to any power desired. e.g. A 1, 3, 1 triangle with addition rule 1A + 3B sums to powers of 4. A 1, 4, 1 triangle with addition rule 1A + 4B sums to powers of 5. etc..

What is actually being addressed on this thread at the moment are the following recursion-based relationships uncovered by Ramsey2879.



I was merely pointing out that the terms involved in his progressions can be derived from the Pell Triangle...

16x + 9 = (4^2)x + 2*4 + 1
144x + 25 = (12^2)x + 2*12 + 1
576x + 49 = (24^2)x + 2*24 + 1
1600x + 81 = (40^2)x + 2*40 + 1
etc.

... and that it ought to be possible to generalize his results to all of the Pell Triangle in some manner as yet TBD.

For instance, Ramsey2879 is working with squares and triangles at the moment. If one were to "step over" one diagonal to the left, might there not perhaps be a way to come up with similar recursion-based formulas for multiples of tetrahedrals (perhaps in polynomial expanded form...) plus some constant that add up to cubes, pentagonal numbers or some other class of square number?

Best,
Raphie
It appears difficult to generalize my result. Although it appears that for N = 4T, where T is a triangular number > 0, there is an infinite sequence of triangular numbers T(x) such that N^{2}*T(x) + 2N + 1 is a square beginning with x = 0,1,a,... The Sophie triangular numbers are the T(x)'s for T = 1. The sequences which I gave are for T = 3 and 6. However, I cannot determine a formula for the sequence of x's for T = 10 and 15 as I could for T = 3 and 6 so I am not so certain that the sequence of T(x)'s is infinite. My results for T = 10 and 15 are below:

40^2 *T(x) + 81 is a square for T(x) = 0,1,488566,6349266,128600703 and 1671258205
60^2 *T(x) + 121 is a square for T(x) = 0,1,743590,11132121,110744403,and 1657929736.

The x's increase in size rapidly and there is no clear recursive formula given only the above.
If you want to try and solve each series, first put each in the order {[6],[4],[2],[1],[3],[5]}.
dodo
#22
Mar23-11, 05:44 PM
P: 688
Sorry for interrupting again with perhaps something obvious, but extending a result on triangular numbers to the other polygonal numbers may not be the proper thing, given that it's only the triangular numbers that appear as a diagonal of Pascal's triangle; subsequent diagonals in Pascal's triangle have no relation (no obvious one, anyway) with the squares, pentagonals, and so on. Why should they?

Edit: Hmm... actually, there IS a relation between polygonal and triangular numbers, given here:
http://en.wikipedia.org/wiki/Polygonal_number#Formulae
namely, the n-th s-gonal number is given by (s-2)*T(n-1)+n. That should lead to relations with the triangle in post #18; (the one which was Pascal's triangle but with diagonals multiplied by powers of 2).

For example, taking the next diagonal of that triangle: 8,32,80,160,... call them R(n), with n=1,2,3... Then
(R(n+1)-R(n))/4+n+2 should produce squares. Examples: (32-8)/4+3=9; (80-32)/4+4=16; (160-80)/4+5=25; (280-160)/4+6=36; (448-280)/4+7=49; ...
Raphie
#23
Mar24-11, 05:42 PM
P: 153
Yes, Dodo, All polygonal numbers are constructible as multiples of Triangular Numbers, plus a Counting Number. e.g. 22 = 3*T_3 + 4; 35 = 3*T_4 + 4; ; 51 = 3*T_5 + 4 --> 4th, 5th and 6th Pentagonal Numbers. Pentagonal Numbers,specifically, also have the property that they are, each and all, 1/3 a Triangular Number.

Here is another way of looking at how Polygonal Numbers are constructed that better demonstrates how they and n-Hedral Numbers are (or can be...) "born" in very similar manner...

Construction Rule: -1A + 2B: Start [1, n]
Number of Summations (i.e. "Overlays"): 1

001,002,003,004,005,006,007,008,009,010... Counting Numbers: Start [1, 1]
001,003,006,010,015,021,028,036,045,055... Triangular Numbers: Start [1, 2]
001,004,009,016,025,036,049,064,081,100... Square Numbers: Start [1, 3]
001,005,012,022,035,051,070,092,117,145... Pentagonal Numbers: Start [1, 4]
001,006,015,028,045,066,091,120,153,190... Hexagonal Numbers: Start [1, 5]
001,007,018,034,055,081,112,148,189,235... Heptagonal Numbers: Start [1, 6]
etc.

e.g. Counting Numbers
[01, 01], 01, 01, 01, 01, 01, 01... -- "Zeroeth" Summation --> (0n + 1)
[01, 02], 03, 04, 05, 06, 07, 08... -- First Summation --> Counting Numbers

e.g. Pentagonal Numbers
[01, 04], 07, 10, 13, 16, 19, 22 -- "Zeroeth" Summation --> (3n + 1)
[01, 05], 12, 22, 35, 51, 70, 92 -- First Summation --> Pentagonal Numbers

compared to...

PASCAL's TRIANGLE/SQUARE
Construction Rule: -1A + 2B: Start [1, 1] or...
Construction Rule: 0A + 1B: Start [1, 1] (amongst others formulations...)
"Number" of Summations: Infinite

001,001,001,001,001,001,001,001,001,001... "Zeroeth" Summation --> (0n + 1)
001,002,003,004,005,006,007,008,009,010... First Summation --> Counting Numbers
001,003,006,010,015,021,028,036,045,055... Second Summation --> Triangular Numbers
001,004,010,020,035,056,084,120,165,220... Third Summation --> Tetrahedral Numbers
etc.

compared to...

PELL NUMBERS
Construction Rule: 1A + 2B: Start [1, 1]
Number of Summations: 1

[1, 1], 3, 07, 17, 041, 099... "Zeroeth" Summation --> (Half Companion Pell Numbers)
[1, 2], 5, 12, 29, 070, 169... First Summation --> (Pell Numbers)
[1, 3], 8, 20, 49, 119, 288... Second Summation --> (see links below...)

Note, for instance, that (41 -1)/2 = 20, (99-1)/2 = 49, etc.
=============================================
Expansion of 1/(1-3x+x^2+x^3).
http://oeis.org/A048739

Or, alternatively...
a(n)-th triangular number is a square: a(n+1) = 6*a(n)-a(n-1)+2, with a(0) = 0, a(1) = 1
If (X,X+1,Z) is a Pythagorean triple, then Z-X-1 and Z+X is in the sequence.
http://oeis.org/A001108

alternating with...
a(n) = 6a(n-1)-a(n-2)+2 with a(0) = 0, a(1) = 3.
Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives X values.
http://oeis.org/A001652



Best,
Raphie
Raphie
#24
Mar31-11, 07:28 PM
P: 153
Quote Quote by ramsey2879 View Post
It appears difficult to generalize my result. Although it appears that for N = 4T, where T is a triangular number > 0, there is an infinite sequence of triangular numbers T(x) such that N^{2}*T(x) + 2N + 1 is a square beginning with x = 0,1,a,... The Sophie triangular numbers are the T(x)'s for T = 1. The sequences which I gave are for T = 3 and 6. However, I cannot determine a formula for the sequence of x's for T = 10 and 15 as I could for T = 3 and 6 so I am not so certain that the sequence of T(x)'s is infinite. My results for T = 10 and 15 are below:

40^2 *T(x) + 81 is a square for T(x) = 0,1,488566,6349266,128600703 and 1671258205
60^2 *T(x) + 121 is a square for T(x) = 0,1,743590,11132121,110744403,and 1657929736.

The x's increase in size rapidly and there is no clear recursive formula given only the above.
If you want to try and solve each series, first put each in the order {[6],[4],[2],[1],[3],[5]}.
The first hint of a pattern I see here is...

128600703/488566 = 263.220738 [5]/[3]
1671258205/6349266 = 263.220694 [6]/[4]

110744403/743590 = 148.932077 [5]/[3]
1657929736/11132121= 148.932062 [6]/[4]

But I don't have enough data to extrapolate further, other than to suggest...

3.38503663 10^10 = (128600703*263.220694)
1.64933923 10^10 = (110744403*148.932062)

... as likely lower bounds for the next number in each series.

- RF
Raphie
#25
Apr8-11, 12:43 PM
P: 153
Ramsey2879, here is a secondary observation that clearly (and cleanly) relates the first differences of the square number indices associated with the 2nd, 3rd and 6th terms of the series you posted -- for 1600(T_x) + 81 and 3600(T_x) + 121 are square -- to the Pell Numbers...

sqrt (3600*1+121) - sqrt (1600*1+81) = (61 - 41) = 20 = 2* 10
sqrt (3600*743590+121) - sqrt (1600*488566+81) = (51739 - 27959) = 23780 = 2378 * 10
sqrt (3600*1657929736+121) - sqrt (1600*1671258205+81) = (2443061 - 1635241) = 807820 = 80782 * 10

A000129 Pell Numbers
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149
http://oeis.org/A000129

Presumably, a second pattern TBD will link the 1st, 4th and 5th terms to the Pell numbers as well.

sqrt (3600*0+121) - sqrt (1600*0+81) = (11 - 9) = 2
sqrt (3600*11132121+121) - sqrt (1600*6349266+81) = (200189 - 100791) = 99398
sqrt (3600*110744403+121) - sqrt (1600*128600703+81) = (631411 - 453609) = 177802

Best,
Raphie

P.S. Data-wise, below are the index numbers of the Triangular Numbers associated with the two series of squares.

(sqrt ((8*0) + 1) - 1) = 0
(sqrt ((8*1) + 1) - 1) = 1
(sqrt ((8*488566) + 1) - 1) = 1976
(sqrt ((8*6349266) + 1) - 1) = 7126
(sqrt ((8*128600703) + 1) - 1) = 32074
(sqrt ((8*1671258205) + 1) - 1)/2 = 57814

(sqrt ((8*0) + 1) - 1) = 0
(sqrt ((8*1) + 1) - 1) = 1
(sqrt ((8*743590) + 1) - 1) = 2438
(sqrt ((8*11132121) + 1) - 1) = 9436
(sqrt ((8*110744403) + 1) - 1) = 29764
(sqrt ((8*1657929736) + 1) - 1) = 115166
ramsey2879
#26
Jun23-11, 02:23 PM
P: 894
Quote Quote by Raphie View Post
Limiting ourselves to N...

Conjecture:

(sqrt (16x + 9) - 1)/2 = y | 2y^2 + 2y - 3 = z^2


for x = a Sophie Germain Triangular Number, which is recursively defined as:
a(n)=34a(n-2)-a(n-4)+11

First 11 values (I have not checked further as I would be very surprised if this equivalency did not hold...)

x = 0, 1, 10, 45, 351, 1540, 11935, 52326, 405450, 1777555, 13773376...

y = 1, 2, 6, 13, 37, 78, 218, 457, 1273, 2666, 7422...

Seems to me there ought to be a (at least a somewhat...) simple algebraic proof for this, if accurate, and my reasoned guess is that Pell Numbers would come in to play somewhere, but observation, research, and heuristically-based approaches to mathematics, not proofs, are my forte.

RELATED PROGRESSIONS:

Sloane's A124174
Sophie Germain triangular numbers tr: 2*tr+1 is also a triangular number.
http://oeis.org/A124174

Sloane's A124124
Nonnegative integers n such that 2n^2+2n-3 is square.
http://oeis.org/A124124
The proof goes as follows:
A) 2T(n) + 1 = n^2 + n + 1 = (y^2+y)/2
B) 16T(n) + 9 = 4y^2 + 4y^2 + 1 (Multiply A by 8 and add 1)
C) 4n^2 + 4n + 1 = 2y^2 + 2y -3 (Mulyiply A by 4 and subtract 3)

Equation B shows that 16T(n) + 9 is (2y+1)^2 from which you derived your y series.
Equation C shows that 2y^2 + 2y -3 is (2n+1)^2
QED
I recently revisited this problem and the proof just stared at my face.
You posted a lot of added comments re Sloanes sequences A124124 and 124174 to which I have a lot to add. I invite you to coauthor some edits to the comments on these sequences with me.


Register to reply

Related Discussions
Proof of Goldbach,Polignac,Legendre,Sophie Germain conjecture.pdf Linear & Abstract Algebra 2
Germain primes Linear & Abstract Algebra 4
Proof of Golbach's conjecture and the twin prime conjecture Linear & Abstract Algebra 9
Find three different right-angled triangles Introductory Physics Homework 17