Factoring

Computer Science Level 4

Let $x$ and $y$ be prime numbers such that both $x! + 1$ and $y! + 1$ are prime numbers as well. If the product $(x! + 1)(y! + 1)$ is a 93-digit long number as shown below,

460432316581264925060046564429192334437963296857959039928362776676511162585133246054400000001

Find $x + y$ .

The answer is 78.

1 solution

Giorgio Coniglio
Jul 28, 2016

Looking at the product of the two primes:

460432316581264925060046564429192334437963296857959039928362776676511162585133246054400000001 we find their factors to be: 13763753091226345046315979581580902400000001 and 33452526613163807108170062053440751665152000000001

and since they are a factorial plus 1 each factorial should be:

13763753091226345046315979581580902400000000 and 33452526613163807108170062053440751665152000000000

By looking at the number of zeros at the end of each factorial, we see they must be in the range of

30! = 265252859812191058636308480000000 and 45!=119622220865480194561963161495657715064383733760000000000

and the two primes between them are 37 and 41 whose factorial plus 1 match the factors above.

So the answer is s = 37 + 41 = 78.

Note: The factoring of this big number can be achieved online here: https://www.alpertron.com.ar/ECM.HTM

Factoring

The answer is 78.

1 solution

0 pending reports