You show several computations. They all seem to work. So what computations are NOT working as desired? It is really unclear what you are asking.

y = (((pnp^2/ x ) + x^2) / pnp)

pnp = x * y

(((((pnp^2 / x) + x^2)) / x) / pnp) where y = 0

If all of these are true in the factors we wish to find, x and y, is there a limit; a range; that could be computed that said if x is this big then y is that big? It wouldn’t be a differential equation that solves a spring. But how do I find and x that is true by testing if y is also true in these 4 constraints?

It is a simple idea, but what is the math that completes it? I know that where y on the graph equals zero x has the value approximate to the smaller factor. I have an equation that will tell me y factor knowing x. If you move x larger y gets smaller. Move x smaller y gets larger. There is only a certain range that will prove this true. Combine that with all the other constraints you have and equation that solves a polynomial.

Well I am confident that these equations will find the magnitude of the smaller factor of the SemiPrime. You find where the graph equals zero and test to see if the x value before zero is the factor. (I know that testing from largest x value before the y value equals zero can also be computationally intensive.)

I can not get it to work with a 128bit SemiPrime because the calculation is too large. I get the error that pnp is not a machine size real number.

Is there any way to analyze the graph to find where it equals zero visually without trial and error?

Thanks Daniel. Sorry for my late reply.

The computations work, but are the goal of the computations correct?

I mean it looks good and all, but what I am trying to do is factor large RSA numbers. The last one was a 2048 bit one. Mathematica gave me the values that would be close to zero in my equation. I claim that the smaller semiPrime factor will occur on the graph of the equation where y equals zero.

So did Mathematica give a correct number where the equation equals zero? And if that number is correct does my hypothesis of where the smaller SemiPrime factor occurs hold true? How do I use Mathematica to confirm?

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I need advice on how to crunch numbers with Mathematica. Someone helped me before and used the monitor class. I want to see the process as the computer computation occurs. In the console Mathematica will indicate processing in the menu bar, but I need assurance it is doing something and how far along the process is.

In my posts, this one and previous, I have tried brute force the semiPrimes smallest factor after I estimate where it occurs. I will provide and example here as a notebook. I have criticism that it is trivial, that my attempts are futile because 10^37 is still to large to crunch. But I took the error of the equation I used to estimate the value and I feel it is close and that it is useful because it significantly reduced the number of values to try.

Either way I want to crunch large numbers. Anyone have any tips or resources?

4.08615746000780408833983218761558688141378368019671197 × 10^37

Factor an unfactored RSA number by estimating the smaller factor on the graph then brute forcing the sample values.

On the graph as x approaches zero, x is the smaller Prime factor. But there is a margin of error so we test values between 5 and zero. From left to right and with any luck we will find the smaller Prime factor.

Refer to the first graph of the attached notebook for the estimate.

https://community.wolfram.com/groups/-/m/t/3066471

If you graph

Clear[x, pnp]    
pnp = 2564855351    
Show[
 Plot[(( (((pnp^2/x) + x^2)) / x) /pnp), {x, 0, 60000}]      
 ]

As y approaches zero from the origin, x approaches the smaller SemiPrime.

Not usually faster than brute force, but it is a graphical representation of what is happening with the SemiPrime.

I am having trouble analyzing the graphs with Mathematica. Is there an add-on that will let me evaluate graphs. Single pictures of graphs can be deceiving.