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We show that any factorization of any composite Fermat number $F_{m} = {2^{2}}^{m} + 1$ into two nontrivial factors can be expressed in the form $F_{m} = (k 2^{n} + 1) (ℓ 2^{n} + 1)$ for some odd $k$ and $ℓ, k \geq 3, ℓ \geq 3$ , and integer $n \geq m + 2, 3 n < 2^{m}$ . We prove that the greatest common divisor of $k$ and $ℓ$ is 1, $k + ℓ \equiv 0 m o d 2^{n}, m a x (k, ℓ) \geq F_{m - 2}$ , and either $3 | k$ or $3 | ℓ$ , i.e., $3 h 2^{m + 2} + 1 | F_{m}$ for an integer $h \geq 1$ . Factorizations of $F_{m}$ into more than two factors are investigated as well. In particular, we prove that if $F_{m} = {(k 2^{n} + 1)}^{2} (ℓ 2^{j} + 1)$ then $j = n + 1, 3 | ℓ$ and $5 | ℓ$ .

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Displaying 61 – 80 of 1807

A New Condition for Consecutive Primitive Roots of a Prime.

A new Fibonacci-like sequence of composite numbers.

A new formulation of the law of octic reciprocity for primes =+-3 (mod 8) and its consequences.

A New Infinite Product Representation for Real Numbers.

A new kind of Diophantine equations

A new method in the elementary theory of primes

A new modulo computation algorithm

A new necessary condition for moduli of non-natural irreducible disjoint covering system.

A note on a paper by Brenner.

A note on character sums over finite fields.

A note on Deaconescu's result concerning Lehmer's problem.

A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$

A Note on Formulae for the nth Prime.

A note on generalized mobius ...-functions

A note on linear recurrent Mahler numbers.

A note on multiplicatively e-perfect numbers.

A note on perfect totient numbers.

A Note on prime k-th power nonresidues.

A note on primes p with $σ (p^{m}) = z^{n}$

A note on pseudoprimes with respect to abelian linear recurring sequence

Currently displaying 61 – 80 of 1807