We obtain all positive integer solutions（m1,m2,a,b） with a ＆gt; b,gcd（a,b） = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z ...We obtain all positive integer solutions（m1,m2,a,b） with a ＆gt; b,gcd（a,b） = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z such that k ＆gt; 0,l ＆gt; 0,r ＆gt; 0,t1 ＆gt; 0,t2 0,gcd（k,l） = 1,and k is square-free.展开更多
In this paper,the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013.Let p be an odd prime and let a∈Z^(+).Then,if p≡1(mod 3),[5/6p^(a)]∑k=0(2kk)/16^(k)≡(3/p^(a))(mod p^(2))is obtained,where(■...In this paper,the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013.Let p be an odd prime and let a∈Z^(+).Then,if p≡1(mod 3),[5/6p^(a)]∑k=0(2kk)/16^(k)≡(3/p^(a))(mod p^(2))is obtained,where(■)is the Jacobi symbol.展开更多
We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squarefree....We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squarefree. Here we remove all these restrictions and prove （similar to the best known one with restrictions） cx2/log X with an absolute constant c that the number of such pair of integers upto a large real X is asymptotic to which we give explicitly. Our error term is also compatible to the best known one.展开更多
基金supported by the Guangdong Provincial Natural Science Foundation (Grant Nos.10152606101000000 and S2012040007653)National Natural Science Foundation of China (Grant No.11271142)
文摘We obtain all positive integer solutions（m1,m2,a,b） with a ＆gt; b,gcd（a,b） = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z such that k ＆gt; 0,l ＆gt; 0,r ＆gt; 0,t1 ＆gt; 0,t2 0,gcd（k,l） = 1,and k is square-free.
基金supported by the National Natural Science Foundation of China(Nos.12001288,12071208)
文摘In this paper,the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013.Let p be an odd prime and let a∈Z^(+).Then,if p≡1(mod 3),[5/6p^(a)]∑k=0(2kk)/16^(k)≡(3/p^(a))(mod p^(2))is obtained,where(■)is the Jacobi symbol.
基金supported by Progetti di Ricerca di Interesse Nazionale 2008“Approssimazione diofantea e teoria algebrica dei numeri”Ministerio de Economía y Competitividad of Spain(Grant No.MTM2015-63829-P)
文摘We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squarefree. Here we remove all these restrictions and prove （similar to the best known one with restrictions） cx2/log X with an absolute constant c that the number of such pair of integers upto a large real X is asymptotic to which we give explicitly. Our error term is also compatible to the best known one.
