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Register a new userOn monochromatic representation of sums of squares of primes
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When the sequences of squares of primes is coloured with $K$ colours, where $K geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) ll K expleft(frac{(3log 2 + {rm o}(1))log K}{log log K}right)$ for $K geq 2$. This improves on $s(K) ll_{epsilon} K^{2 +epsilon}$, which is the best available upper bound for $s(K)$.
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In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,winmathbb{N}={0,1,cdots})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3yin{4^k:kinmathbb{N}}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.
We study sign changes in the sequence ${ A(n) : n = c^2 + d^2 }$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher and Murty, we show that there are at least $X^{frac{1}{4} - epsilon}$ sign changes in each interval $[X, 2X]$ for $X gg 1$. This improves to $X^{frac{1}{2} - epsilon}$ many sign changes assuming the Generalized Lindel{o}f Hypothesis.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 le r le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
The Macaulay2 package SumsOfSquares decomposes polynomials as sums of squares. It is based on methods to rationalize sum-of-squares decompositions due to Parrilo and Peyrl. The package features a data type for sums-of-squares polynomials, support for external semidefinite programming solvers, and optimization over varieties.
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 geq p^theta$ for $theta=1/2+1/2000.$ This improves the work of Matomaki (2009) who obtained the result for $theta=1/2-varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $theta=4/9-varepsilon.$ Similarly as in the work of Matomaki, we apply Harmans sieve method to detect primes $p equiv 1 , (d^2)$. To break the $theta=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as Zhang (2014) used to obtain equidistribution beyond the Bombieri-Vinogradov range for primes with respect to smooth moduli. To optimize the argument we incorporate technical refinements from the Polymath project (2014). Since the moduli are squares, the method produces complete exponential sums modulo squares of primes which are estimated using the results of Cochrane and Zheng (2000).
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