With Concern Numbers A_3n[ Where A_3n=3^2n + 2 And n Is An Integer >=0 ]; And A Glance On Numbers Of The Form k+F_n, Where k in {2,4,8}} And F_n Is a Fermat Number

We denote by A3.n, the number of the form 32^n+2, where n is an integer ≥0; in other words, for every integer n≥0,A3.n=32^n+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 j∈{0,1,2,3},A3.j is prime and A3.4 is composite ( since A3.4≡0mod[19] )]. In this paper, we show [via elementary arithmetic congruences] the following result (T.). For every positive integer n such that n≡4mod[6], we have A3.n-2≡17mod[19]; and for every integer n≥2, we have A3.n-2≡1mod[5]. Result (T.) immediately implies that there are infinitely many composite numbers of the form A3.n. 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.n. A Fermat number is a number of the form F_n=22^n+1, where n is an integer ≥0. A Fermat composite (see Dickson (1952) or Hardy, & Wright, (1979) or Annouk, (2012)) is a non prime Fermat number and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in Annouk, (2012) and in Annouk, (2014). It is known (see [4] ) that for every j∈{0,1,2,3,4},F_j is a Fermat prime (F_0=22^0+1=3. and 3 is prime, F_1=22^1+1=5 and 5 is prime, F_3=22^3+1=257 and 257 is prime, and F_4=22^4+1=65537 and 65537 is prime), and it is also known (see Hardy, & Wright (1979) or Hoffman (2000)or Křížek, et al. (2002) or Keller, (2020) or Euler, (1738)or Scharlau, & Opolka, (1985) that F_5 and F_6 are Fermat composites (F_5=22^5+1=641×6700417., and since 2013, it is known that F₂₇₄₇₄₉₇=2^{2²⁷⁴⁷⁴⁹⁷}+1 is Fermat composite number). Fermat numbers have importance and their application to other sciences such as cryptography, neural networks, electronic computer, polygons with straightedge, filtering, autocorrelation, and related areas with conventional computing are clearly known. Factorization of Fermat numbers (specially for big one) is a very hard problem in number theory and cryptography too. The biggest known Fermat prime is F_4=22^4+1=65537, and this number is used in cryptography because of the fact that this prime helps the cryptography less vulnerable to the public exponent attack lowly (as mentioned "Coppersmith's short pad attack" in the literature). Practical and efficient methods are still in need for factorization of such numbers, even there are many algorithms to factorize some of such composite numbers. That being so, in this paper, we show [via elementary arithmetic congruences] the following result (E.). For every integer n>0 such that n≡1mod[2], we have F_n-1≡4mod[7]; and for every integer n≥2, we have F_n-1≡1mod[j], where j∈{3,5}. Result (E.) immediately implies that there are infinitely many composite numbers of the form 2+F_n. Result (E.) also implies that the only prime of the form 4+F_n is 7 and the only primes of the form 8+F_n 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+F_n.