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Dan11
science forum Guru Wannabe

Joined: 03 Oct 2005
Posts: 125

Posted: Tue Jul 18, 2006 7:05 pm    Post subject: Another look at triangle number factoring.

Another look at triangle number factoring.

The examples below are simple but when it
comes to a --->oo of composites it becomes much more
complex when the ratio of their 2 factors are not close
to (2).
The few examples I give below are simple but -----> oo
in number.
e.g.
Simple factoring of integers ---4141,4171,4183
These are not 0 (mod 3) or 0(mod 5) and all have
just 2 factors of equal length.
Where these integers fall on the same t(n) line
because the ratio of their two factors are very close
to (2) but always < > 2

4141 belongs on t(91) line
t(91) = 4186
4186 - 4141 = 45
45 = t(9)
91 - 9 = 82
82/2 = 41 one of the factors of 4141.

4171 also belongs on t(91) line
t(91) = 4186
4186 - 4171 = 15
15 = t(5)
91 - 5 = 86
86/2 = 43 one of the factors of 4171.

4183 also belongs on t(91) line
t(91)= 4186
4186 - 4183 = 3
3 = t(2)
91 - 2 = 89
89 is one of the factors of 4183

A little more complex where the target composite
is not on the same t(n) line because the ratio of
the two factors of equal length is not close to (2) .
---
4351559 belongs on t(2950) line.
t(2950) = 4352725
4352725 - 4351559 = 1166
The next > t(n) = 1176
1176 - 1166 = 10 = t(4)
10/2 = 5
2950 + 5 = 2955
t(2955) = 4367490
4367490 - 4351559 = 15931 = t(178)
2955 - 178 = 2777
2777 is one of the 2 factors of 4351559.

It appears that if the two prime factors ratio
is not close to 2 then the composite has to be
subtracted from a larger t(n) than the original
line t(n).

A much larger example would be a much larger composite
with just 2 prime factors of equal length where their
factors ratio is very close to (2).

The composite with two prime factors of equal length ---

1.0415234943969403527027914891372177438417149151824
38208839471211168779357685762633676578794750054754
46378985021512468272206349247261722091418225580800
96273209244588701455002402643457425701380486179003
43218501017285565300154340840862741475517390440755
02399021189285642476581348257918150825641338721177
62700563213600336758932533114089414793949204418163
00085249683215403827452982325270761261333989104011
99216099011743793304735767481049033742236990137722
32572832073883857269445567695195346483756738463225
57676866092179413261590086826489887188328152058092
58697160827547371515287970901834768702441909484312
335854411059035533e+617

Where it resides on the same t(n) line # =

4.5640409603704048105123305093096359291499147608293
37293274319128396698693930460579328085738922303300
21053272900193148911206862759791547365284905263144
67077706778351105557309958686574344456236215260461
84614625627662453497231121655886593174713965525345
34643100605866670549402771320469461852661010569248
699995741e+308

And its corresponding t(n) =

1.0415234943969403527027914891372177438417149151824
38208839471211168779357685762633676578794750054754
46378985021512468272206349247261722091418225580800
96273209244588701455002402643457425701380486179003
43218501017285565300154340840862741475517390440755
02399021189285642476581348257918150825641338721177
62700563213600336758932533114089414793949204418163
00085249683215403827452982325270761261333989104011
99216099011743793304735767481049033742236990137722
32572832073883857269445567695195346483756738463225
57676866092179413261590086826489887188328152058092
58697160827547371515287970901834768702441909484312
335854411059067411e+617

Subtracting the ---
t(n) - (the composite with two prime factors) = 31878

31878 = t(252)

Line # =

4.5640409603704048105123305093096359291499147608293
37293274319128396698693930460579328085738922303300
21053272900193148911206862759791547365284905263144
67077706778351105557309958686574344456236215260461
84614625627662453497231121655886593174713965525345
34643100605866670549402771320469461852661010569248
699995741e+308

(-) minus 31878 =

4.5640409603704048105123305093096359291499147608293
37293274319128396698693930460579328085738922303300
21053272900193148911206862759791547365284905263144
67077706778351105557309958686574344456236215260461
84614625627662453497231121655886593174713965525345
34643100605866670549402771320469461852661010569248
699963863e+308

Which is the largest prime factor of the composite.
The composite being the first large integer in this
example.

This is just an exercise in trying to determine other
large composites that have just 2 prime factors of equal
length where the ratio of these factors is NOT close to (2). Thus not on the same t(n) line and much more difficult if not impossible to factor large composites
of this form by using this method.
Most are unsolvable where the ratios are not close
to (2) but quite a few small composites < 10^7 are solvable by manipulating the t(n)!
In other words no closed form for finding these more
difficult ones unlike the composites that are on the
same t(n) line that have a difference of a smaller t(n).

This is probably all trivial but could be a spring board
for furthur study of t(n) - composite = a smaller t(n)
which in turn produces either the larger or smaller
factor in the composite.

Dan
Proginoskes
science forum Guru

Joined: 29 Apr 2005
Posts: 2593

Posted: Thu Jul 20, 2006 6:46 am    Post subject: Re: Another look at triangle number factoring.

Dan wrote:
 Quote: Another look at triangle number factoring. [nice math research clipped, so I could eventually get down to:] This is probably all trivial but could be a spring board for furthur study of t(n) - composite = a smaller t(n) which in turn produces either the larger or smaller factor in the composite.

Remember that t(n) = n(n-1)/2, so if you have the equation

t(n) - C = t(m),

then

C = t(n)-t(m) = (n-m)(n+m-1)/2,

which helps unless n-m = 2 or n+m = 3.

--- Christopher Heckman
Dan11
science forum Guru Wannabe

Joined: 03 Oct 2005
Posts: 125

Posted: Thu Jul 20, 2006 7:29 pm    Post subject: Re: Another look at triangle number factoring.

Dan wrote:
 Quote: Another look at triangle number factoring. [nice math research clipped, so I could eventually get down to:] This is probably all trivial but could be a spring board for further study of t(n) - composite = a smaller t(n) which in turn produces either the larger or smaller factor in the composite. Remember that t(n) = n(n-1)/2, so if you have the >equation

You lost me here, isn't' that supposed to be n(n+1)/2?

 Quote: t(n) - C = t(m), then C = t(n)-t(m) = (n-m)(n+m-1)/2, which helps unless n-m = 2 or n+m = 3. --- Christopher Heckman

Dan

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