Ming-Chit Liu

On generalized quadratic equations in three prime variables

AbstractLeta1,a2 anda3 be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation

Conditional bounds for small prime solutions of linear equations

a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b

On binary equations

is insoluble in primespj. We obtain CardE(X)≤X1-ε. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation

The fractional parts of a sum of polynomials

x_1^2 + x_2^2 + x_3^k = b

THE NUMBER OF POWERS OF 2 IN A REPRESENTATION OF LARGE EVEN INTEGERS II

in integral variablesxj with the above bound for CardE(X) better than that in [2].

Squares of Primes and Powers of 2

AbstractEssentially sharp bounds for small prime solutionspj,qi of the following two different types of equations are obtained.