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1.
Let p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n
Recently, Ono proved for every prime m 5 that there are infinitely many congruences of the form p(An+B)0 (mod m). However, his results are theoretical and do not lead to an effective algorithm for finding such congruences. Here we obtain such an algorithm for primes 13m31 which reveals 76,065 new congruences. 相似文献
2.
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. 相似文献
3.
Shaun Cooper 《The Ramanujan Journal》2002,6(4):469-490
Let r
k(n) denote the number of representations of an integer n as a sum of k squares. We prove that
where
Here n = 2 p
p
p
is the prime factorisation of n, n is the square-free part of n, the products are taken over the odd primes p, and (
) is the Legendre symbol.Some similar formulas for r
7(n) and r
9(n) are also proved. 相似文献
4.
LetN be a sufficiently large even integer and
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5.
Limit theorems for the ratio of the empirical distribution function to the true distribution function 总被引:4,自引:0,他引:4
Jon A. Wellner 《Probability Theory and Related Fields》1978,45(1):73-88
Summary We consider almost sure limit theorems for
and
where
n
is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda
n
0. It is shown that (1) ifna
n
/log2
n then both
and
converge to 1 a.s.; (2) ifna
n
/log2
n=d>0 (d>1) then
has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna
n
/log2
n0 then
and
have limit superior + almost surely. Similar results are established for the inverse function
n
–1
.Supported by the National Science Foundation under MCS 77-02255 相似文献
6.
Yoshitaka Uda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1995,46(3):366-383
We study nonnegative solutions of the initial value problem for a weakly coupled system
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