Please post code, not images.

The plot is accurate, the expression value is highly oscillatory and negative in the plotted domain.

ClearAll[eq1, x];
eq1 = -2564855351 + 2564855351 Sqrt[2564855351] x Sqrt[1/(x + 2564855351 x^2)]

values = Table[{x, N@eq1}, {x, 41223, 41281}]
values = Table[{x, N@eq1}, {x, 41223, 41281, .05}]

Extend the domain

Plot[eq1, {x, 1000, 100000}, PlotRange -> All]

The expression is never >=0, its limit is 0 as x -> Infinity

Limit[eq1, x -> Infinity]

Thanks for all your help. Are you guessing and checking or is there a function that tells where graphs intersect.

In the Pappy Craylar Conjecture (the last two graphs I posted) the smaller Prime factor occurs when y is between zero and one.

My question is: With the number of values between one and zero. Is the Pappy Craylar Conjecture useful. Obviously Mathematica knows the factor and can factor it in seconds. But if N was larger than what Mathematica could factor would the PCC be useful?

For and N of 2564855351, testing odd values between 50,000 and 41,227 is around 4,000 values. (Note that in the Mathematica Notebooks, I called “N” pnp to prevent conflict.) I can plug in a known x with pnp and it checks, but I cannot solve the equation with knowing pnp alone, because it results in imaginary number values. But are there equations that show the relationships in Prime numbers like the PCC equations do?

And thanks again

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This is the corrected equations. Should meet at 41,000 (the Prime factor). Comes within 0.2 units. The problem is that more than just the Prime factor come close. I would have to test from 50,000 to 41,000. I would have to test 4,500 values. Less than 10^9. But still large.

Thanks everyone that was extremely helpful. I fixed the one equation. But I was crunching numbers to see why I couldn’t get the graph to intersect. Your corrections say it all. I am getting the two separate graphs to come to 0.2, close at 41,000. But they do not intersect. Is there any class or method that can evaluate when they “almost” intersect?

This is supposed to factor SemiPrimes. PNP is the semi-Prime and x is the factor.

Also in Print[ii], ii is not defined, should probably be Print[i], and the parenthesis are not balanced. With i = 3 the test provided to While fails because of the < condition, perhaps you meant <=.

Why won't this loop?

clear [i, pnp]

pnp= 2564855351


i=3; while[  ((((pnp^2/i + i^2) / pnp * i) – (i^3/pnp)) << pnp, Print[ii]; i=i+2]

I'm not sure I entirely understand what you're asking for, but if you want to find where two functions intersect, you might just try using Solve:

f[N_] := x^4/(N^2 + x);
g[N_] := (N^2/x + x^2)/N*x - N;
Solve[f[10] == g[10], x]
(* {{x -> 0}, {x -> 0}, {x -> 0}, {x -> 100/9}} *)

With that you could determine where to focus your plot, or maybe add some visualization that highlights the intersections.

Can you be more specific about what views you want? I guess I don't know what a "real time" graph is. I also don't know what "I need that tools on a regular..." means.