With Concern Numbers A₃n Where A₃n=3²n + 2 And n Is An Integer >=0 ; And A Glance On Numbers Of The Form k+Fₙ, Where k in 2, 4, 8} And Fₙ Is a Fermat Number

We denote by A₃. ₍, the number of the form 3^2ⁿ+2, where n is an integer 0; in other words, for every integer n0, A₃. ₍=3^2ⁿ+2 it is easy to check that A₃. ₀=5, A₃. ₁=11, A₃. ₂=83, A₃. ₃=6563 and A₃. ₄=43046723; moreover, for every j0, 1, 2, 3, A₃. ₉ is prime and A₃. ₄ is composite (since A₃. ₄0mod[19) ]. In this paper, we show via elementary arithmetic congruences the following result (T. ). For every positive integer n such that n4mod6, we have A₃. ₍-217mod19; and for every integer n2, we have A₃. ₍-21mod5. Result (T. ) immediately implies that there are infinitely many composite numbers of the form A₃. ₍. That being so, using result (T. ) coupled with a special case of a Theorem of Dirichlet on arithmetic progressions, we explain why it is natural to conjecture that there are infinitely many prime numbers of the form A₃. ₍. A Fermat number is a number of the form Fₙ=2^2ⁿ+1, where n is an integer 0. A Fermat composite (see Dickson (1952) or Hardy, and for every integer n2, we have Fₙ-11modj, where j3, 5. Result (E. ) immediately implies that there are infinitely many composite numbers of the form 2+Fₙ. Result (E. ) also implies that the only prime of the form 4+Fₙ is 7 and the only primes of the form 8+Fₙ are twin primes 11 and 13. That being said, using result (E. ) and a special case of a Theorem of Dirichlet, we conjecture that there are infinitely many primes of the form 2+Fₙ.