  Prime numbers, whose properties are important subjects in mathematics, are
also fundamental in computer science notably in IT security, Cryptocurrencies
as Bitcoin and Blockchain, cryptography, Code theory notably Error detection
codes, integer factorization, and random number generation. Finding prime
numbers is too an active area of research in mathematics. There are many
methods for identifying and generating them and many primality tests which are
often complex and expensive in terms of calculation time, and many conjectures
and theorems related to prime numbers, such as the prime number theorem,
Goldbach's conjecture, and the Riemann hypothesis. My objective in this work is
to propose two conjectures :
  $1)$ the integer
  $1+3.2^{20n}$ is not a prime number for all $n=0$ or $1$ mod $3$, and $2)$
the integer
  $1+3.2^{4+20n}$ is not a prime number for all $n=2$ mod $3$.
  Keywords : Prime numbers; IT security; Cryptography.
