I. INTRODUCTION Let a set of s-distinct symbols be arranged in a s×s square in such a way that every symbol occurs exactly once in every row and every column. Such a square is called a Latin square of order s. Tw...I. INTRODUCTION Let a set of s-distinct symbols be arranged in a s×s square in such a way that every symbol occurs exactly once in every row and every column. Such a square is called a Latin square of order s. Two Latin squares are called orthogonal if, when one of them is superposed on the other, every symbol of the first square occurs with every symbol of the second one once and only once. Now we denote by N(s)展开更多
In his work on Waring’s problem for cubes and biquadrates,Brüdern has established the following: Almost all natural numbers can be expressed as the sum of three cubes and a biquadrate of natural numbers. This is...In his work on Waring’s problem for cubes and biquadrates,Brüdern has established the following: Almost all natural numbers can be expressed as the sum of three cubes and a biquadrate of natural numbers. This is comparable with Davenport’s result that almost all natural numbers are the sum of four cubes of natural numbers. We observe that one can apply Vaughan’s p-adic iterative technique to problem展开更多
In the present report, we further improve the pruning technique in the Hardy-Littlewood method and advance a new method by which Vaughan’s p-adic iterative technique is applied to the problem of sums of mixed powers....In the present report, we further improve the pruning technique in the Hardy-Littlewood method and advance a new method by which Vaughan’s p-adic iterative technique is applied to the problem of sums of mixed powers.We suppose that ε is a sufficiently small positive number, η=η(ε)is a small positive number depending on ε at best and that n is sufficiently large in terms of ε and η.展开更多
After K. F. Roth had proved that almost all integers might be expressed as the sum of a square, a cube and a fourth power of positive integers, in 1952 K. Prachar obtained that almost all positive even integers n may ...After K. F. Roth had proved that almost all integers might be expressed as the sum of a square, a cube and a fourth power of positive integers, in 1952 K. Prachar obtained that almost all positive even integers n may be represented展开更多
Let v(n) denote the number of representations of n as the sum of six cubes and twobiquadrates of natural numbers. Then for all sufficiently large n, we have v(n)≥1/(48)~2(log16/15)Γ(4/3)~6Γ(5/4)~2?(Γ(5/2))?(n)n^(3...Let v(n) denote the number of representations of n as the sum of six cubes and twobiquadrates of natural numbers. Then for all sufficiently large n, we have v(n)≥1/(48)~2(log16/15)Γ(4/3)~6Γ(5/4)~2?(Γ(5/2))?(n)n^(3/2),where ?(n) =sum from q=1 to ∞ sum from a=1 (a,q)=1 to q q^(-8)S_3(q,a)~6S_4(q,a)~2c(-an/q) S_k(q,a) =sum from r=1 to q e(ar^k/q).展开更多
Let r(n) denote the number of representations of n as the sums of 5 cubes and 3 biquadra-tes of natural numbers. Then for all sufficiently large n, one has r(n)(?)n17/12, which is the expected order of magnitude of r(n).
It is shown that almost all natural numbers can be expressed as the sum of three cubes aud one fifth power of natural numbers. To be more precise, we have E(N)<<N^(1-19/2163+),where E(N) is the number of natural...It is shown that almost all natural numbers can be expressed as the sum of three cubes aud one fifth power of natural numbers. To be more precise, we have E(N)<<N^(1-19/2163+),where E(N) is the number of natural numbers not exceeding N and not being the sum of three cubes and one fifth power.展开更多
LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_...LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP 2 is an almost prime with at most two prime factors.展开更多
文摘I. INTRODUCTION Let a set of s-distinct symbols be arranged in a s×s square in such a way that every symbol occurs exactly once in every row and every column. Such a square is called a Latin square of order s. Two Latin squares are called orthogonal if, when one of them is superposed on the other, every symbol of the first square occurs with every symbol of the second one once and only once. Now we denote by N(s)
基金Project supported by the National Natural Science Foundation of China.
文摘In his work on Waring’s problem for cubes and biquadrates,Brüdern has established the following: Almost all natural numbers can be expressed as the sum of three cubes and a biquadrate of natural numbers. This is comparable with Davenport’s result that almost all natural numbers are the sum of four cubes of natural numbers. We observe that one can apply Vaughan’s p-adic iterative technique to problem
基金Project supported by the National Natural Science Foundation of China.
文摘In the present report, we further improve the pruning technique in the Hardy-Littlewood method and advance a new method by which Vaughan’s p-adic iterative technique is applied to the problem of sums of mixed powers.We suppose that ε is a sufficiently small positive number, η=η(ε)is a small positive number depending on ε at best and that n is sufficiently large in terms of ε and η.
文摘After K. F. Roth had proved that almost all integers might be expressed as the sum of a square, a cube and a fourth power of positive integers, in 1952 K. Prachar obtained that almost all positive even integers n may be represented
基金Project supported by the Alexander von Humboldt-Fund and the National Natutal Science Foundation of China.
文摘Let v(n) denote the number of representations of n as the sum of six cubes and twobiquadrates of natural numbers. Then for all sufficiently large n, we have v(n)≥1/(48)~2(log16/15)Γ(4/3)~6Γ(5/4)~2?(Γ(5/2))?(n)n^(3/2),where ?(n) =sum from q=1 to ∞ sum from a=1 (a,q)=1 to q q^(-8)S_3(q,a)~6S_4(q,a)~2c(-an/q) S_k(q,a) =sum from r=1 to q e(ar^k/q).
基金Project supported by the National Natural Science Foundation of China.
文摘Let r(n) denote the number of representations of n as the sums of 5 cubes and 3 biquadra-tes of natural numbers. Then for all sufficiently large n, one has r(n)(?)n17/12, which is the expected order of magnitude of r(n).
基金Project supported by the National Natural science Foundation of China
文摘It is shown that almost all natural numbers can be expressed as the sum of three cubes aud one fifth power of natural numbers. To be more precise, we have E(N)<<N^(1-19/2163+),where E(N) is the number of natural numbers not exceeding N and not being the sum of three cubes and one fifth power.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 19531010 and 19801021)
文摘LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP 2 is an almost prime with at most two prime factors.
