Related papers: 9 Divides no Odd Fibonacci

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely…

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4…

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which…

We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for…

In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary…

Let $\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge…

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard…

In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.

Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced and investigated. Abelian n-torsion clean rings are somewhat characterized and a complete characterization of strongly n-torsion clean rings…

In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.

In this paper, I proved that $$N=p_1+p_2+2p_3, p_1\sim N/2, p_2\sim N/2, p_3=o(N),$$ where $N$ is a large even number, and $p_i\ (i=1,2,3)$ are odd primes.

A proof that there is no $3 \times 3$ magic square constructed with nine distinct square numbers is given.

There has been always an ambiguity in division when zero is present in the denominator. So far this ambiguity has been neglected by assuming that division by zero as a non-allowed operation. In this paper, I have derived the new set of…

We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which…

The sequence $F_{dn+h}$ and its convolutions have (for $h=0$) been studied in a recent paper at the arxiv [arXiv:2603.08636]. The instance with general $h$ is more involved and uses Chebyshev polynomials.

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. The main result is twofold: (1) we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the…

In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.

We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this…

We use probability urn models to discover some known and unknown series identities involving Fibonacci numbers.