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A NEW FORM OF DISCRIMINANT FOR HILL'S EQUATION   总被引：4，自引：0，他引：4

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We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.  相似文献

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NOTES ON GLAISHER''''S CONGRUENCES   总被引：1，自引：0，他引：1
Let p be an odd prime and let n ≥ 1,k ≥ 0 and r be integers. Denote by Bk the kth Bernoulli number. It is proved that (i) If r≥1 is odd and suppose p≥r+4, thenp-1∑j=11/(np+j)=-(2n+1+(x+1)/2(x+2)Bp-r-2p2(modp3).(ii)IFr≥2is even and suppose p≥r+3,thenp-1∑j=11/(np+j)+=r/r+1Bp-x-1p(modp2).(iii)p-1∑j=11/(np+j)p-2=-(2n+1)p(modp2).Thisesult generalizes the Glaisher's congruence. As a corollary, a generalization of the Wolstenholme's theorem is obtained.  相似文献

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The population evolution equations are theoretical bases of the research on population problems. It is the following boundary value problem: p(0,t)=v(t) in (0,T), (1) p(r,0) =u(r) in Ω= (0,r_m) , where t denotes time. r denotes age, r_m is the highest age ever attained by individual of the population, p(r,t)=N(r,t)/r is called age density of the population,  相似文献

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1·引理记△ABC的三边分别为a,b,c,其内切圆、外接圆半径分别为r,R,p=12(a b c),则tanA2=rp-a,tanB2=rp-b,tanC2=rp-c.2·推证关于tanA2,tanB2,tanC2的等式.由tanA2=rp-a=rp-2RsinA=rp-2R·2tanA21 tan2A2,得ptan3A2-(4R r)tan2A2 ptanA2-r=0,记f(x)=px3-(4R r)x2 px-r,则tanA2为f(x)=0的根.同样tanB2,tanC2亦为f(x)=0的根.∴tanA2 tanB2 tanC2=4R rp,tanA2tanB2 tanB2tanC2 tanC2tanA2=1,tanA2tanB2tanC2=rp,简记为∑tanA2=4R rp,∑tanA2tanB2=1,∏tanA2=rp.3·推证有关cotA2,cotB2,cotC2的恒等式.由1cotA2=rp-2RsinA…  相似文献