THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

Correa may yet win fourth term in Ecuador

Subject Pre-election politics in Ecuador. Significance Deteriorating economic conditions, declining public spending and falling support for the government have provided opposition forces with a favourable climate to make gains in advance of next year's general elections. However, with little over eight months before voters are scheduled to go to the polls, the opposition is fragmented and the main challengers are uncertain. The political landscape is further complicated by uncertainty over who will stand for the ruling party. While President Rafael Correa has repeatedly stated that he will not compete, he may yet seek election for a fourth successive term. Impacts Constitutional reform, media freedom, security and tax reductions will be the focus of electoral campaigns from the right and centre. Preventing large-scale mining, environmentalism, creating a plurinational state and wealth redistribution will be central to the left. The full list of parties and candidates authorised to compete in the elections will not be known until the year-end.

Divided opposition boosts Lukashenka's re-election bid

Significance The failure of opposition parties to agree a joint candidate -- and calls for an election boycott by some -- pave the way for President Alexander Lukashenko to win a fifth successive term. Impacts For now, Lukashenka will remain beholden to Moscow as his main source of economic aid and political support. However, Belarus will increasingly court Chinese and South Asian investment to lessen reliance on Russia. Lukashenka's election campaign will stress need for continuity and highlight Belarus as a bastion of stability compared to Ukraine.

II. On fermat’s theorems of the polygonal numbers

This paper relates to the second theorem, viz. that which asserts that every number is composed of 4 square numbers (0 [or zero] being considered as an even square). If every odd number be composed of 4 square numbers, then every even number must also be composed of 4 square numbers; for every even number must, on a continued division by 2, ultimately become an odd number. The paper relates chiefly to the Table which accompanies it, from which it appears that a remarkable law obtains as to the division of odd numbers (2 n + 1) into 4 square numbers—when a number of the form 4 n + 1 is divisible into 2 square numbers, which (as 4 n + 1 is an odd number) must be one of them odd, the other even. Before explaining the Table, it is proper to state that if an odd number be divisible into 4 square numbers, three of them must be odd, and one of them even, or one of them must be odd, and 3 of them even, otherwise their sum cannot be an odd number; it follows from this that if the difference between any two of them be an odd number, the difference between the other two must be an even number, and vice versâ ; for let a 2 + b 2 + c 2 + d 2 = 2 n + 1, then if a 2 - b 2 = 2 p , c 2 - d 2 must equal 2 q + 1; if possible let c 2 — d 2 = 2 r , then a 2 - b 2 + c 2 - d 2 = 2 p +2 r ; add 2 b 2 + 2 d 2 (an even number) to each, and a 2 + b 2 + c 2 + d 2 will be an even number, which by the hypothesis it is n o t; if, therefore, a 2 — b 2 be an even number, c 2 — d 2 cannot also be an even number, and therefore must be an odd one. If, therefore, the four roots of the squares into which any odd number may be divided are arranged in any order there will be three differences; the two exterior differences will be one odd, the other even; the middle difference may be either odd or even. The Table is arranged thus:—the lowest row of figures is the series 1, 5, 9, 13, 17, &c. (4 n + 1); the next row above is the series of natural numbers, 0, 1, 2, 3, 4, &c. ( n ), &c.; the next row is 1, 3, 5, 7, 9, &c. (2 n + 1) the odd numbers; each of the odd numbers is the first term in a series increasing upwards by the numbers 2, 4, 6, 8, 10, &c., forming an arithmetic series of the second order (the first and second differences being respectively 2 each); when the number in the lowest row cannot be divided into 2 squares the arithmetic series is not formed, and the squares are marked with an asterisk, but when the number 4 n + 1 is divisible into 2 square numbers, the roots of these squares constitute the two exterior differences of the roots into which the odd number may be divided, and also of the roots into which each term of the series increasing upward may be divided; the middle difference of the roots will be the smaller half of the sum of the 2 roots of the square numbers into which 4 n + 1 may be divided, with a negative sign, and will increase by 1 in each successive term of the upward series.

On the Summation of a Compound Series, and its application to a Problem in Probabilities.

The series proposed for summation isIn which series each line or term is the product of two factorials, the first consisting ofp, the last ofqfactors of successive numbers. And in each successive term the factors of the first factorial are diminished each by unity, and the factors of the last increased.

On the Summation of a Compound Series, and its application to a Problem in Probabilities

The series proposed for solution in the following paper is—The law of this series is manifest. Each term is the product of two factorials—the first consisting of p, and the latter of q, factors; and in each successive term, the factors of the first factorial are each diminished by one. and those of the latter increased by one.

XXXIV.—Summation of a Compound Series, and its Application to a Problem in Probabilities

The series propossed for solution in the follwing paper is—The law of this series is manifest. Each term is the product of two factorials, the first consisting of p, and the latter of q factors. And in each successive term, the factors of the first factorial are each diminished by one, and those of the latter increased by one.