Factoring RSA weak modulus Given a public key for RSA, I have extracted the modulus which looks like this :

Public-Key: (2049 bit)
Modulus:
    01:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:02:19:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:fe:
    f8:45
Exponent: 65537 (0x10001)


If we remove the first 01 for the sign, we have a 2048 modulus and we can remove the zeros. So I guess the modulus will be :

219FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEF845


My problem now is to factorize this 1036 bit modulus. I think that the FF pattern gives us a hint but I can't find it.

Could you please help? 

Thank you
As fgrieu points out, the number to factor can be written as 
$$
n = 2^{2048} + 538\cdot 2^{1024} - 67515\,.
$$
We can identify this number with the polynomial $f(x) = x^2 + 538x - 67515$, 
 with $n = f(2^{1024})$. Furthermore, if we factor this polynomial as $f(x) = g(x)\cdot h(x)$ then $n = f(2^{1024}) = g(2^{1024})\cdot h(2^{1024}) = p\cdot q$ since polynomial evaluation is homomorphic.

Since this is a simple quadratic equation, it is easy enough to find that $f(x) = (x - 105)(x + 643)$. Thus, the factors are $2^{1024} - 105$ and $2^{1024} + 643$.

The Fermat factorization algorithm is particularly efficient for integers whose factors are very close together. This happens to be the case here.