Pretty Large

Number Theory Level 3

Since \( x ^ { ab } - 1 = (x^a -1 ) \times \left( x^{ a (b-1) } + x^ { a (b-2) } + \cdots + x^ a + 1 \right) \), we conclude that the number \( 2^ {105} -1 \) has factors of \( 2^3 - 1 = 7 \), \( 2^5 - 1 = 31 \) and \( 2^{7} - 1 = 127 \). Is \[ \frac{ 2 ^ { 105} - 1 } { \big( 2^3 -1 \big) \times \big( 2^5 -1 \big) \times \big(2 ^ 7 -1 \big) } \] prime?