L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
First 50 documents
Young Modules and Schur Subalgebras - Henke (1999) (Correct)
....to p in the sense of James if and only if Q i2N 0 Gamma [n] i [m] i Delta = 1. The definition of p contained as given in definition 1.1.8 is used further below to reformulate James s description of the decomposition numbers for two part partitions. Lemma 1.1.9 (E. Lucas,1890; see Dickson [7]) Let p be a prime and let n; m be non negative integers. Then Gamma n m Delta j Q i2N 0 Gamma [n] i [m] i Delta modulo p. Let n; m be non negative integers. By lemma 1.1.9 we have Gamma n m Delta j Q i2N 0 Gamma [n] i [m] i Delta modulo p. Then Gamma n m Delta j ....
L.E. Dickson, History of the theory of numbers, vol. 1, NewYork: Chelsea, 1952.
Solving Elliptic Diophantine Equations by estimating Linear.. - Tzanakis (1994) (3 citations) (Correct)
....,i.e. we describe a general practical method for computing explicitly all integral solutions of equations of the form V 2 = Q(U) where Q(U) 2 Z[U ] is a quartic polynomial with non zero discriminant. For such equations one can nd in the litterature only sporadic results; see Chapter XXII of [Di] and sections D24 of [V] and [G] Very few of them are of a general character and these deal with special types of quartic elliptic equations (like,for example,x 4 Dy 2 = 1) Most of the results consist in solving completely speci c numerical examples by clever but often very laborious ad hoc ....
L.E. Dickson,\History of the Theory of Numbers", Vol. II,Chelsea Publ. Co.,New York, 1971.
Fermat's Last Theorem - Darmon, Diamond, Taylor (2000) (1 citation) (Correct)
....Euler wrote down a proof of Fermat s Last Theorem for the exponent = 3, by performing what in modern language we would call a 3 descent on the curve x 3 y 3 = 1 which is also an elliptic curve. Euler s argument (which seems to have contained a gap) is explained in [Edw] ch. 2, and [Dic1], p. 545. It took mathematicians almost 100 years after Euler s achievement to handle the case = 5; this was settled, more or less simultaneously, by Gustav 3 Peter Lejeune Dirichlet [Dir] and Adrien Marie Legendre [Leg] in 1825. Their elementary arguments are quite involved. cf. Edw] sec. ....
L.E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea Publ. Co., New York, 1971.
When Newton met Diophantus: A study of rational-derived.. - Buchholz, MacDougall (1999) (1 citation) (Correct)
....so that A : a b c, B : a b c, C : a b c, then the second derivative constraint becomes 3x 2 = A 2 B 2 C 2 . Notice that (A, B, C, x) 1, 1, 1, 1) is a particular solution and since this is a homogeneous quadratic we can use the chord method (mentioned in [12]) to parametrize all solutions as dA = u 2 v 2 w 2 2uv 2uw, dB = u 2 v 2 w 2 2uv 2vw, dC = u 2 v 2 w 2 2uw 2vw, dx = u 2 v 2 w 2 . 2) where d = gcd(r.h.s # s) and u, v, w # Z. By solving these equations for a, b, c we obtain ....
L. E. Dickson, History of the Theory of Numbers, vol. 2, Chelsea, (1952).
Some Arithmetic Properties Of Shanks's Generalized Euler And.. - Teske, Williams (Correct)
.... 1, then 2n 1 = m(p Gamma 1) 2 and m is odd; thus, T 2n ( j (p Gamma1) 2 X k=1 Gammap k k ( p Gamma1) 2)m Gamma1 (mod p) j (p Gamma1) 2 X k=1 Gammap k k p 1 k (mod p) j (p Gamma1) 2 X k=1 1 k (mod p) It is a result going back to Eisenstein (see Dickson [8], p.105, p.109) that (2 p Gamma1 Gamma 1) p j Gamma2 (p Gamma1) 2 X k=1 1 k (mod p) We now have the following results. Theorem 5.6. If p j Gamma1 (mod 4) and (p Gamma 1) 2 j 2n 1, then p j c p;n if and only if 2 p Gamma1 j 1 (mod p 2 ) Corollary 5.6.1. If p j Gamma1 (mod ....
E. Dickson, History of the theory of numbers, vol. 1, Chelsea, New York, 1952.
A Report On Wiles' Cambridge Lectures - Rubin, Silverberg (1994) (2 citations) (Correct)
....found and the problem remained open, inspiring many generations of mathematicians. Much of modern number theory has been built upon attempts to prove Fermat s Last Theorem. For details on the history of Fermat s Last Theorem (last because it is the last of Fermat s questions to be answered) see [5], 6] and [26] 1991 Mathematics Subject Classification. Primary 11G05; Secondary 11D41, 11G18. The authors thank the NSF for financial support. 1 2 K. RUBIN AND A. SILVERBERG What Andrew Wiles announced in Cambridge was that he could prove many elliptic curves are modular, sufficiently ....
....by ae E;5 , and let S be the set of cusps of X . Then X is a curve defined over Q which has the following properties. 20 K. RUBIN AND A. SILVERBERG ffl The rational points on X Gamma S correspond to isomorphism classes of pairs (E 0 ; OE) where E 0 is an elliptic curve over Q and OE : E[5] E 0 [5] is a GQ module isomorphism. ffl As a complex manifold X Gamma S is four copies of H= Gamma(5) so each component of X has genus zero. Let X 0 be the component of X containing the rational point corresponding to (E; identity) Then X 0 is a curve of genus zero defined over Q ....
[Article contains additional citation context not shown here]
Dickson, L. E., History of the theory of numbers (Vol. II), Chelsea Publ. Co., New York (1971).
Third-Degree Diophantine Equations - Garaev (Correct)
....Equations M. Z. Garaev Received February, 1997 1. INTRODUCTION This work is devoted to the investigation of the solvability of the Diophantine equation x 3 y 3 z 3 = nxyz; 1) where n = 2 f0; 3g, n is an integer. The rst results were obtained as long ago as 1856 by Sylvester (see [1]) He proved that for n = 6 (1) had no solutions in nonzero integers. The next results were obtained almost eighty years later. In 1933 34 Hurwitz and Mordell showed that for n = 1 and for n = 5 (1) had the only solutions (1; 1; 1) and (1; 1; 2) respectively, with an accuracy to within the ....
....no solutions in the positive integers x, y, z. Corollary. For the values of n indicated in Theorem 1 (2) has no solutions in the positive integers x, y, z. In order to prove the theorem, we shall need some auxiliary statements which we shall formulate as lemmas. Lemma 1 belongs to Sylvester (see [1]) Lemma 1. Let A, B, C, D, be arbitrary real numbers for which A 3 B 3 C 3 = D : 94 THIRD DEGREE DIOPHANTINE EQUATIONS 95 Then ABCf 3 g 3 h 3 = Dfgh; where h = A 2 B 6 3 B 2 C 6 3 C 2 A 6 3 3ABC 3 3 3 ; g = AB 2 ....
Dickson, L. E., History of the Theory of Numbers, vol. 2, New York, 1934.
A note on Shanks's Chains of Primes - Teske, Williams (Correct)
....2a 1 = 0 in IF q , then p(x) has the zero Gammaa=3 with multiplicity 3; furthermore, if x = Gammaa=3, then g(x; s) x. Thus, we exclude this possibility and we find that (D=q) Gamma3=q) j q (mod 3) By classical results concerning the solubility of cubic congruences modulo q (see Dickson [5], p. 256) we know that h(x) can have 3 zeros in IF q if and only if q j (D=q) mod 3) and ff (q 2 Gamma1) 3 j 1 (mod q) where ff = A p D) 2, A = 16as 2a Gamma 20s 13. We note that ff = 4s Gamma 2a 1)fl, where fl is a zero of x 2 Gamma (4a Gamma 3)x 4s Gamma 2a 1. If i is a ....
E. Dickson, History of the theory of numbers, vol. 1, Chelsea, New York, 1952.
An Extension of an Old Problem of Diophantus and Euler. II - Dujella (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966.
An Extension of an Old Problem of Diophantus and Euler - Dujella (1999) (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966, pp. 518--519.
Unknown - (Correct)
No context found.
Dickson, L.E., History of the Theory of Numbers, Vol. 2. Chelsea, New York, 1966, pp. 513--520.
An Extension of an Old Problem of Diophantus and Euler. II - Dujella (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966.
An Extension of an Old Problem of Diophantus and Euler - Dujella (1999) (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966, pp. 518--519.
A note on Diophantine quintuples - Dujella (2000) (Correct)
No context found.
Dickson, L.E., History of the Theory of Numbers, Vol. 2. Chelsea, New York, 1966, pp. 513--520.
Last revised: 30 December 2003 - Congruences For Fibonacci (Correct)
No context found.
L.E. Dickson, History of the Theory of Numbers, Vol.I, Chelsea, New York, 1952, pp. 393-407.
An extension of an old problem of Diophantus and Euler - Dujella (1999) (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966, pp. 518--519.
An Extension of an Old Problem of Diophantus and Euler. II - Dujella (2002) (Correct)
No context found.
L. E. Dickson. History of the Theory of Numbers, Vol. 2, New York: Chelsea, 1966.
Fibonacci and Galois Representations of Feedback-With-Carry.. - Goresky, Klapper (2002) (1 citation) (Correct)
No context found.
L. E. Dickson, History of the Theory of Numbers. Washington, DC: Carnegie Inst., 1919.
A generalization of a second irreducibility theorem of I. Schur - Martha Allen And (Correct)
No context found.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, New York, 1971, 744. 14
On the Number of Representations of Certain Integers as Sums of.. - Cooper (2002) (Correct)
No context found.
L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952.
An Infinite Set Of Heron Triangles With Two Rational Medians - Buchholz, Rathbun (1997) (Correct)
No context found.
Dickson, Leonard E. History of the Theory of Numbers, volume 2, Chelsea, 1952.
On Triangles With Rational Altitudes, Angle Bisectors Or Medians - Buchholz (1999) (Correct)
No context found.
L.E. Dickson, History of the Theory of Numbers, vol. 2, Chelsea, 1952.
Perfect Pyramids - Buchholz (1992) (Correct)
No context found.
L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, 1952.
Some Formulae For Partitions Into Squares - Hirschhorn (2000) (Correct)
No context found.
L. E. Dickson, History of the Theory of Numbers, Chelsea, New York, 1966.
Prime-Producing Cubic Polynomials - Mott, ROSE (Correct)
No context found.
L.E. Dickson, History of the Theory of Numbers, reprinted by Chelsea 1952.
First 50 documents
Online articles have much greater impact More about CiteSeer.IST Add search form to your site Submit documents Feedback
CiteSeer.IST - Copyright Penn State and NEC