"New" same approach that isn't factorization

[Alberico Lepore]

before understanding this

Originally Posted by retina

Well go on then, have at it.

I have to understand this

Originally Posted by Alberico Lepore

What characteristic must N have for that equality to be true

for example for N = 121

this

2 * 121 + 2 * 1 ^ 2 + y ^ 2- (22) ^ 2 = 0

it's not true

that is, y is not integer

would you give me a little clue

[retina]

Originally Posted by Alberico Lepore

I have to understand this

would you give me a little clue

That is your equation, you invented it. So you should be the one to understand it. I have no idea what you are trying to do, it all looks like nonsense to me.

It's up to you to show us how you factor things, not the other way around. Your claim, you prove it.

[Alberico Lepore]

Originally Posted by retina

That is your equation, you invented it. So you should be the one to understand it. I have no idea what you are trying to do, it all looks like nonsense to me.

It's up to you to show us how you factor things, not the other way around. Your claim, you prove it.

So maybe I found when it's true:
(a + b) mod 3 = 0
in two cases

M=[(a+b)/2-((a+b)/6-1)/2]*[2*[(a+b)/2-((a+b)/6-1)/2]-3]
and
M=[(a+b)/2-((a+b)/6+1)/2]*[2*[(a+b)/2-((a+b)/6+1)/2]+3]



so I tried to bring back a generic number (a + b) mod 3 = 0

in

M=[(a+b)/2-((a+b)/6-1)/2]*[2*[(a+b)/2-((a+b)/6-1)/2]-3]

but I didn't get any useful results

Example

N=161
,
2*(N+(n/2)^2-((a+b)/6-1)^2)+2*a^2+((b-a)/2)^2=((3*a+b)/2)^2
,
a*b=(N+(n/2)^2-((a+b)/6-1)^2)
,
2*(N+(n/2)^2-((a+b)/6-1)^2)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0

but I will continue to study

[Alberico Lepore]

Bruteforce could be attempted for a multiple of 9 :
9*F

N=161
,
2*(N*9*F)+2*a^2+((b-a)/2)^2=((3*a+b)/2)^2
,
a*b=(N*9*F)
,
2*(N*9*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0
,
F=15

->
a=105

GCD(105,161)=7


but I think this is very RANDOM

[Alberico Lepore]

If we solve F as a function of a and N

solve 2*(N*9*F)+2*a^2+((b-a)/2)^2=((3*a+b)/2)^2 , a*b=(N*9*F) , 2*(N*9*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0 ,F,b

->

9*N*F=2*a^2-3*a

multiplying by 2 and imposing 2 * a = A

we will have 18*N*F=A^2-3*A

A0 < sqrt(18*N)

is it possible to apply the Coppersmith method?

https://en.wikipedia.org/wiki/Coppersmith_method

[mathwiz]

Please, stop posting.

[Alberico Lepore]

Originally Posted by Alberico Lepore

If we solve F as a function of a and N

solve 2*(N*9*F)+2*a^2+((b-a)/2)^2=((3*a+b)/2)^2 , a*b=(N*9*F) , 2*(N*9*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0 ,F,b

->

9*N*F=2*a^2-3*a

multiplying by 2 and imposing 2 * a = A

we will have 18*N*F=A^2-3*A

A0 < sqrt(18*N)

is it possible to apply the Coppersmith method?

https://en.wikipedia.org/wiki/Coppersmith_method

I have found other equations where, perhaps, the Coppersmith method is applicable

I don't know with what efficiency

solve (N*F-1)/8=(X^2-1)/8-2*((b-a)/8)^2 ,a*b=(N*F) , 2*(N*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0

8*X^2-6*X-9=F*N*9

8*X^2+6*X-9=F*N*9

multiplying everything by 2 and imposing A = 4 * X and B = 4 * X

are obtained

A^2-3*A-18=F*N*9*2

B^2+3*B-18=F*N*9*2



solve (65*F-1)/8=(X^2-1)/8-2*((b-a)/8)^2 ,a*b=(65*F) , 2*(65*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0

8*X^2-6*X-9=F*65*9

8*X^2+6*X-9=F*65*9

multiplying everything by 2 and imposing A = 4 * X and B = 4 * X

are obtained

A^2-3*A-18=F*65*9*2

B^2+3*B-18=F*65*9*2


and these are the first two

and then

solve (N*F-1)/8=x*(x+1)/2-2*((b-a)/8)^2 ,a*b=(N*F) , 2*(N*F)+2*1^2+((a+b)/2+1)^2-((3*a+b)/2)^2=0 ,F

32*x^2+20*x-7=F*N*9

32*x^2+44*x+5=F*N*9

multiplying everything by 2 and imposing A = 8 * X and B = 8 * X

are obtained

A^2+5*A-14=F*N*9*2

B^2+11*B+10=F*N*9*2

and these are the other 2

[CRGreathouse]

Originally Posted by Alberico Lepore

is it possible to apply the Coppersmith method?

Sure, if you have enough information about the factor. If you don't know anything about the factors then it's much slower than other methods.

[Alberico Lepore]

Originally Posted by CRGreathouse

Sure, if you have enough information about the factor. If you don't know anything about the factors then it's much slower than other methods.

What kind of information about the factors?

[Alberico Lepore]

Originally Posted by Alberico Lepore

A^2-3*A-18=F*N*9*2

B^2+3*B-18=F*N*9*2



A^2+5*A-14=F*N*9*2

B^2+11*B+10=F*N*9*2

and these are the other 2

Infinite equations of this type can be generated.

[Dr Sardonicus]

Originally Posted by Alberico Lepore

Infinite equations of this type can be generated.

I am brought to mind of the following:

We crossed a Walk to the other Part of the Academy, where, as I have already said, the Projectors in speculative Learning resided.

The first Professor I saw was in a very large Room, with forty Pupils about him. After Salutation, observing me to look earnestly upon a Frame, which took up the greatest part of both the Length and Breadth of the Room, he said perhaps I might wonder to see him employed in a Project for improving speculative Knowledge by practical and mechanical Operations. But the World would soon be sensible of its Usefulness, and he flattered himself that a more noble exalted Thought never sprung in any other Man's Head. Every one knew how laborious the usual Method is of attaining to Arts and Sciences; whereas by his Contrivance, the most ignorant Person at a reasonable Charge, and with a little bodily Labour, may write Books in Philosophy, Poetry, Politicks, Law, Mathematicks and Theology, without the least Assistance from Genius or Study. He then led me to the Frame, about the Sides whereof all his Pupils stood in Ranks. It was twenty Foot Square, placed in the middle of the Room. The Superficies was composed of several bits of Wood, about the bigness of a Dye, but some larger than others. They were all linked together by slender Wires. These bits of Wood were covered on every Square with Paper pasted on them, and on these Papers were written all the Words of their Language, in their several Moods, Tenses, and Declensions, but without any Order. The Professor then desired me to observe, for he was going to set his Engine at Work. The Pupils at his Command took each of them hold of an Iron Handle, whereof there were fourty fixed round the Edges of the Frame, and giving them a sudden turn, the whole Disposition of the Words was entirely changed. He then commanded six and thirty of the Lads to read the several Lines softly as they appeared upon the Frame; and where they found three or four Words together that might make part of a Sentence, they dictated to the four remaining Boys who were Scribes. This Work was repeated three or four Times, and at every turn the Engine was so contrived that the Words shifted into new Places, as the Square bits of Wood moved upside down.

Six Hours a-day the young Students were employed in this Labour, and the Professor shewed me several Volumes in large Folio already collected, of broken Sentences, which he intended to piece together, and out of those rich Materials to give the World a compleat Body of all Arts and Sciences; which however might be still improved, and much expedited, if the Publick would raise a Fund for making and employing five hundred such Frames in Lagado, and oblige the Managers to contribute in common their several Collections.

He assured me, that this Invention had employed all his Thoughts from his Youth, that he had emptyed the whole Vocabulary into his Frame, and made the strictest Computation of the general Proportion there is in Books between the Numbers of Particles, Nouns, and Verbs, and other Parts of Speech.

I made my humblest Acknowledgments to this illustrious Person for his great Communicativeness, and promised if ever I had the good Fortune to return to my Native Country, that I would do him Justice, as the sole Inventer of this wonderful Machine; the Form and Contrivance of which I desired Leave to delineate upon Paper, as in the Figure here annexed. I told him, although it were the Custom of our Learned in Europe to steal Inventions from each other, who had thereby at least this Advantage, that it became a Controversy which was the right Owner, yet I would take such Caution, that he should have the Honour entire without a Rival.

-- from Chapter 5 of Gulliver's Travels by Jonathan Swift

But an infinity of nonsense! You've got the Professor at the grand Academy of Lagado beat, hands down.