I gave it in my post.

Would you show the two-step factorization of 23999887836455503939, please?

Is this one?

With the new algorithm any semiprime number could factorized almost directly.

Here is the new general algorithm to factorize odd semiprime numbers.

n=731

1. Enter n=731

2. Enter A=2n=1462

3. Compute k and r such as A=k^2+r where k^2 is the biggest square less than A. A=38^2+18

Hence k=38 and r=18

4. Compute k-r=38-18=20

5. 20 is of the form s(s+1) where s=4

4*5=20

if not of the form of s(s+1) then go the algo "non composite" number

6. Compute c1=s^2+r=5^2+18=16+18=34

7. Compute c2=(s+1)^2+r=25+18=43

8. Compute gcd(n,c1)=gcd(731,34)=17

9. Compute gcd(n,c2)=gcd(731,43)=43

10. Check 731=17*43 print the factors p=17 q=43 end

Now we have factorized a number which is composite = product of 2 consecutive elements of (k,18) class.

Algo "non composite"number :

Assume that k is < r we have to find an A=n*2m such as r < k

This the first problem to solve :

We try some numbers m such as r become < k

Or we find a general solution to this problem.

let A=2m*n with r<k

If k-r is not of the form s(s+1)

Example : n=129 A=2*129=258

k=16 r=2

r<k

k-r=16-2=14

14 is not of the s(s+1)

We have to find a square t^2 such as :

tk-t^2*r=s(s+1)

t=3

3*16-9*2=30

30 is of the s(s+1)=5*6

Hence 258*9=2322

We enter a new A = 258*9=2322 and we start at step 2

The number is now a composite one and we are sure of that hence :

After all the steps :

k=48 r=18

c1=5^2+18=43

c2=6^2+18=54

n=129=43*3 done

.....................................

A number is direct hit or not.

I mean either a number is COMPOSITE in its class (k,r) and it will be easy to be factorized.

Either it is not then we have to "tranform" it on COMPOSITE in another class.

It will be then easily factorized.

That is the core of my algorithm.

I did it SUCCESSFULLY with 5-6 digits numbers.

I do not know what are the hurdles if the size of n get bigger.

It is up to you to check it and tell me the problems ahead.

If no particular problem then the factorization is over.

Thank you