We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient p...We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.展开更多
0 The Diophantine equation X^(2p)-Dy^2=1Let D be a positive integer which is square free,and p be a prime.In 1966,Ljunggren showed that if p=2 and D=q is a prime,then the Diophantine equationx^(2p)-Dy^2=1(1)has only p...0 The Diophantine equation X^(2p)-Dy^2=1Let D be a positive integer which is square free,and p be a prime.In 1966,Ljunggren showed that if p=2 and D=q is a prime,then the Diophantine equationx^(2p)-Dy^2=1(1)has only positive integer solutions(q,x,y)=(5,3,4),(29,99,1820).In 1979,KoChao and Sun Qi showed that if p=2 and D=2q,then Eq.(1)has no positive inte-展开更多
文摘We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.
文摘0 The Diophantine equation X^(2p)-Dy^2=1Let D be a positive integer which is square free,and p be a prime.In 1966,Ljunggren showed that if p=2 and D=q is a prime,then the Diophantine equationx^(2p)-Dy^2=1(1)has only positive integer solutions(q,x,y)=(5,3,4),(29,99,1820).In 1979,KoChao and Sun Qi showed that if p=2 and D=2q,then Eq.(1)has no positive inte-
