When $p$ and $q$ Are Close

Suppose that $p$ and $q$ are ``close'' to each other. Then it is easy to factor $n$ using a factorization method of Fermat.

Suppose $n=pq$ with $p>q$, say. Then

$$n = \left(\frac{p+q}{2}\right)^2 - \left(\frac{p-q}{2}\right)^2.$$

Since $p$ and $q$ are ``close'',

$$s = \frac{p-q}{2}$$

is small,

$$t = \frac{p+q}{2}$$

is only slightly larger than $\sqrt{n}$, and $t^2-n=s^2$ is a perfect square. So we just try

$$t =$$   ceil$$(\sqrt{n}), \quad t=$$ceil$$(\sqrt{n})+1, \quad t=$$ceil$$(\sqrt{n})+2, \ldots$$

until $t^2-n$ is a perfect square $s^2$. Then

$$p = t+s,\qquad q=t-s.$$

Example 2.1   Suppose $n=23360947609$. Then

$$\sqrt{n} = 152842.88\ldots.$$

If $t=152843$, then $\sqrt{t^2-n} = 187.18\ldots$.

If $t=152844$, then $\sqrt{t^2-n} = 583.71\ldots$.

If $t=152845$, then $\sqrt{t^2-n} = 804\in\mathbb{Z}$.

Thus $s=804$. We find that $p=t+s=153649$ and $q=t-s=152041$.

tex2html_preform ? n=23360947609 %1 = 23360947609 ? sqrt(n) %2 = 152842.8853725288712694157797 ? x=%2 %3 = 152842.8853725288712694157797 ? floor(x+1) %4 = 152843 ? t=floor(x+1) %5 = 152843 ? t^2-n %6 = 35040 ? sqrt(t^2-n) %7 = 187.1897433087614445431082470 ? t++ %8 = 152844 ? sqrt(t^2-n) %9 = 583.7182539547654063924081356 ? t++ %10 = 152845 ? sqrt(t^2-n) %11 = 804.0000000000000000000000000 ? s=804 %12 = 804 ? p=t+s %13 = 153649 ? q=t-s %14 = 152041 ? p*q %15 = 23360947609 ? n %16 = 23360947609 ? factor(n) %17 = [152041 1] [153649 1]

Here is a bigger example in PARI:

? q=nextprime(random(10^50))
%20 = 78177096444230804504075122792410749354743712880803
? p=nextprime(q+1)  \\ a nearby prime
%21 = 78177096444230804504075122792410749354743712880899
? n=p*q
%22 = 6111658408450564697085634201845976850509908580949986889525704...
      ...259650342157399279163289651693722481897
? t=floor(sqrt(n))+1
  ***   precision loss in truncation
? \p150        \\ set precision of floating-point computations.
   realprecision = 154 significant digits (150 digits displayed)
? t=floor(sqrt(n))+1
%29 = 78177096444230804504075122792410749354743712880851
? sqrt(t^2-n)
%30 = 48.000000000000000000000000000000000000000000000000000000....
? s=48
%31 = 48
? t + s     \\ p
%33 = 78177096444230804504075122792410749354743712880899
? t - s     \\ q
%35 = 78177096444230804504075122792410749354743712880803