Related papers: Sums of Two Squares Visualized
We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We…
We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.
Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…
In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number $p$ as a sum of two squares, given a solution of…
We use a variation of the Circle Method, along with the Saddle Point Method, to obtain an asymptotic formula for the number of partitions of a number n into integers which are sums of two squares. Unlike previous work on partitions into…
This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.
We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization…
This paper gives an algorithm to determine whether a number in a cyclic quartic field is a sum of two squares, mainly based on local-global principle of isotropy of quadratic forms.
We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.
A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m^2 and 4m(m+1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's…
In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…
By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
We prove that every integer greater than two may be written as the sum of a prime and a square-free number.
Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares…
One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.