Leo Alex/Lorraine Foster: On the diophantine equation 1+x+y=z.
Rocky Mountain J. Math. 22 (1992), 11-62.
1755 P. Alexandrov (ed.): Die Hilbertschen Probleme. Leipzig 1983.
22778 Avner Ash/Robert Gross: Fearless symmetry. Princeton UP 2006, 270p. Eur 14.
4588 P. Bachmann:Das Fermatproblem in seiner bisherigen Entwicklung.
Springer 1976.
7434 Alan Baker: Review of the book "Catalan's conjecture" by Paulo Ribenboim.
Bull. AMS 32 (1995), 110-112.
27508 Edward Barbeau: Power play. MAA 1997, 190p. Eur 36.
17767 Yuri Bilu: Catalan without logarithmic forms. Internet 2004, 11p.
1990 S. Bloch: The proof of the Mordell conjecture.
Math. Intell. 6/2 (1984), 41-47.
19050 Enrico Bombieri/Walter Gubler: Heights in diophantine geometry.
Cambridge UP 2006, 650p. Eur 110. Well worth the price.
Enrico Bombieri/J. Vaaler: On Siegel's lemma.
Inv. Math. 73 (1983), 11-32.
Jonathan Borwein/Petr Lisonek: Applications of integer relation
algorithms. Preprint 1997.
[Try http://www.cecm.sfu.ca/preprints/1997pp.html]
7041 Nigel Boston/Andrew Granville: Review of the book "The world's
most famous math problem" by Marilyn vos Savant.
Am. Math. Monthly 102 (1995), ...
A. Bremner/J. Cassels: On the equation y#2=x(x#2+p).
Math. Comp. 42 (1984), 257-264.
23310 Jörg Brüdern/R. Dietmann/J. Liu/T. Wooley: A Birch-Goldbach theorem.
Arch. Math. 94 (2010), 53-58.
25183 Mihai Cipu: Gröbner bases and diophantine analysis.
J. Symb. Comp. 43 (2008), 681-687.
2043 T. Cochrane: The diophantine equation f(x)=g(x).
Proc. AMS 109 (1990), 573-577.
11593 Henri Cohen: Algorithmes remarquables en theorie des nombres.
Internet ca. 1997, 14p.
20976 Henri Cohen: Number theory I. Tools and diophantine equations.
Springer 2007, 650p. Eur 54.
D. Cox: Primes of the form x^2+ny^2. Fermat, class field theory and complex
multiplication. Wiley 1989, 350p. Pds. 35.
Jeanine Daems: A cyclotomic proof of Catalan's conjecture. Internet 2003, 65p.
17768 Jeanine Daems: Het vermoeden van Catalan.
Nieuw Arch. Wisk. September 2004, 221-225.
Harold Davenport: Analytic methods for diophantine equations and diophantine
inequalities. 1962.
1863 M. Davis: Hilbert's tenth problem is unsolvable.
Am. Math. Monthly 80 (1973), 233-269.
P. Epstein: Zur Auflo''sbarkeit der Gleichung x^2-Dy^2=-1.
J. Math. 171 (1934), ...
18167 Kurt Girstmair: Kroneckers Lösung der Pellschen Gleichung auf dem Computer.
Math. Semesterber. 53 (2006), 45-64.
6013 Fernando Gouvea: A marvelous proof.
Am. Math. Monthly 101 (1994), 203-222.
A very clear introduction to the mathematical ideas in Wiles's proof
of Fermat's conjecture.
Hans Grauert: Mordells Vermutung ueber Punkte auf algebraischen Kurven
und Funktionkoerper. Publ. IHES 25 (1965), 363-381.
6955 Barry Green/Florian Pop/Peter Roquette: On Rumely's local-global
principle. Jber. DMV 97 (1995), 43-74.
26332 Yves Hellegouarch: Invitation aux mathematiques de Fermat-Wiles.
Dunod 2009, 380p. Eur 40.
5363 John Horgan: Fermat's MacGuffin.
Scientific American September 1993, 14-15.
In June 1993 Andrew Wiles proposed a proof of Fermat's last theorem,
although the complete paper, 200 pages long, has still to be examined in
detail, most experts believe the proof should be true. For seven years,
after that Frey and Ribet had reduced the problem to a (difficult!)
problem about elliptic curves, Wiles virtually stopped writing papers,
attending conferences or even reading anything unrelated to his goal.
6016 Allyn Jackson: Update on proof of Fermat's last theorem.
Notices AMS 41 (1994), 185-186.
18813 Günter Köhler/Jürgen Spilker: Endziffernblöcke von Potenzen.
Math. Sember. 54 (2007), 95-102.
25245 Angel Kumchev: Review of the book "Rational number theory in the 20th century"
by W. Narkiewicz. MR 3014464.
1887 Serge Lang: Some theorems and conjectures in diophantine equations.
Bull. AMS 66 (1960), 240-249.
1883 Serge Lang: Review of Mordell's "Diophantine equations".
Bull. AMS 76 (1970), 1230-1234.
1890 Serge Lang: Higher dimensional diophantine problems.
Bull. AMS 80 (1974), 779-787.
M. Le: A note on the number of solutions of the generalized Ramanujan-
Nagell equation x^2-D=k^n. Acta Arithm. 78 (1996), 11-18.
27097 John Leech: The rational cuboid revisited.
Am. Math. Monthly 84/7 (1977), 518-533.
H. Lenstra: Solving the Pell equation. Notices AMS February 2002, 182-192.
25774 Hendrik Lenstra/Peter Stevenhagen: Review of the book "Solving the Pell
equation" by M. Jacobson/H. Williams. Bull. AMS ... (2014), 7p.
23728 Aimeric Malter/Dierk Schleicher/Don Zagier: New looks on old number theory.
Am. Math. Monthly 120/3 (2013), 243-264.
R. Mason: On Thue's equation over function fields.
J. London Math. Soc. 24 (1981), 414-426.
17866 Yuri Matiyasevich: Hilbert's tenth problem. What can we do with
diophantine analysis? Internet 1996, 26p.
27328 Barry Mazur: Questions about powers of numbers.
Notices AMS February 2000, 195-202.
17766 Tauno Metsänkylä: Catalan's conjecture - another old diophantine problem
solved. Bull. AMS 41/1 (2003), 43-57.
Preda Mihailescu: Reflection, Bernoulli numbers and the proof of Catalan's conjecture.
ECM 2004, 325-340.
Preda Mihailescu: Primary cyclotomic units and a proof of Catalan's conjecture.
J. reine angew. Math. 572 (2004), 167-195.
25492 Preda Mihailescu: Around ABC. EMS Newsletter September 2014, 29-34.
22426 Richard Mollin: Advanced number theory with applications.
CRC Press 2009, 470p. Eur 67.
1920 L. Mordell: Review of Lang's "Diophantine geometry".
Bull. AMS 70 (1964), 491-498.
L. Mordell: Diophantine equations. Academic Press 1969.
L. Mordell: On the rational solutions of the indeterminate equations
of the third and fourth degree. Proc. Camb. Phil. Soc. 21 (1922), 179-192.
21038 Wladyslaw Narkiewicz: Rational number theory in the 20th century.
Springer 2012, 650p. Eur 105.
1888 A. Ogg: Diophantine equations and modular forms.
Bull. AMS 81 (1975), 14-27.
9761 Attila Pethö: On the solution of the equation G_n=P(x).
In 9745 Philippou/Bergum/Horadam, 193-201.
10378 Christoph Po''ppe: Die Fermatsche Vermutung ist bewiesen -
nun auch offiziell. Spektrum 1997/8, 113-116.
Andrew Wiles erha''lt den Wolfskehlpreis.
Alf van der Poorten: Notes on Fermat's last theorem. Wiley 1996. $44.
L. Redei: U''ber die Pellsche Gleichung t^2-du^2=-1.
J. Math. 173 (1935), ...
L. Redei: Die 2-Ringklassengruppe des quadratischen Zahlko''rpers und
die Theorie der Pellschen Gleichung.
Acta Math. Ac. Sci. Hung. 4 (1953), ...
1891 Paulo Ribenboim: Recent results on Fermat's last theorem.
Can. Math. Bull. 20 (1977), 229-242. [3323]
27457 Paulo Ribenboim: 13 lectures on Fermat's last theorem. Springer 1979, 160p.
Paulo Ribenboim: Catalan's conjecture.
Academic Press 1994, 360p. 0-12-587170-8. $65.
Catalan's equation is x^p-y^q=1.
5641 Kenneth Ribet: Wiles proves Taniyama's conjecture; Fermat's last
theorem follows. Notices AMS 40 (1993), 575-576.
Kenneth Ribet: On the equation a^p+2^\alfa b^p+c^p=0.
Acta Arithm. 74 (1997), 7-16.
27463 Karl Rubin: The solving of Fermat's last theorem. Talk UC Irvine 2007, 46.
6177 Karl Rubin/A. Silverberg: A report on Wiles' Cambridge lectures.
Bull. AMS 31 (1994), 15-38.
5990 Marilyn vos Savant: The world's most famous math problem.
St. Martin's Press 1993. 0-312-10657-2. 80p. Pds. 12.
This is not a book at all, but a set of notes, it contains practically
nothing original, it deals with the nature of mathematics mostly quoting
around (typically you read about Goedel what in 1984 the "Scientific
American" wrote about him). It has no depth. Mathematically this is no
surprise, since the author is not a professional mathematician. But the
delusion is that the author is also philosophically so weak. Whatever it
is, it has nothing to do with Wiles's proof. The author writes rather
authoritatively about the nature of proofs in general, but what weight
can one give to her arguments, when one reads the following statement?
"But with contradictions inherent in the mathematical system used in a
proof, how can one ever really prove anything by contradiction? Imaginary
numbers are one example. The square root of +1 is a real number
because +1 x +1 = +1; however, the square root of -1 is imaginary because
-1 times -1 would also equal +1, instead of -1. This appears to be a
contradiction. Yet it is accepted, and imaginary numbers are used
routinely. But how can we justify using them to prove a contradiction?"
And the following sentence in her booklet is fatal for any mathematical
ambitions on her side, certainly not for Wiles: "A possible fatal flaw
in Wiles's proof is whether the same basic arguments could be constructed
to hold true for all exponents, instead of just the exponents equal to or
greater than 3. If it could, the same proof would "prove" the Pythagorean
theorem [...] to be false." On pages 68-69 she proposes four new
mathematical tasks, the last one I cannot understand (one should prove
the parallel postulate, in order to demolish Einstein's theory of
relativity, but until that page she accepted non-Euclidean geometries?)
Well, the "book" includes, on pages 73-76, in the price a transcription
in readable mathematical fonts of Rubin's Internet announcement. (je)
1852 Wolfgang Schmidt: Analytische Methoden fuer diophantische Gleichungen.
Birkhaeuser 1984.
3595 Wolfgang Schmidt: Diophantine approximations and diophantine equations.
SLN Math. 1467 (1991).
8584 Rene' Schoof: Fermat's last theorem.
In 8567 Beutelspacher/, 193-211.
6459 Manfred Schroeder: How probable is Fermat's last theorem?
Math. Intell. 16/4 (1994), 19-20.
Ernst Selmer: The diophantine equation ax{3}+by{3}+cz{3}=0.
Acta Math. 85 (1951), 203-362. A very important paper.
11731 Simon Singh/Kenneth Ribet: Die Lo''sung des Fermatschen Ra''tsels.
Spektrum 1998/1, 96-103.
7267 Vladimir Sprindzhuk: Classical diophantine equations.
SLN Math. 1559 (1993), 230p. 3-540-57359-3. DM 52.
26129 Jörn Steuding: Diophantine analysis. Internet 2003, 79p.
14796 Ian Stewart: Die Rinder des Sonnengottes.
Spektrum 2001/1, 114-115. Two amusing exemples of Pell's equation.
Richard Taylor/Andrew Wiles: Ring theoretic properties of certain
Hecke algebras. Manuscript, october 1994.
26840 Robert Tichy: Abelpreis für Andrew Wiles. IMN 232 (2016), 19-24.
Ilan Vardi: Archimedes' cattle problem.
Am. Math. Monthly 105 (1998), 305-...
18144 Otmar Venjakob: Können zeta-Funktionen diophantische Gleichungen lösen?
DMV-Mitteilungen 14/3 (2006), 136-141. Nützliche kleine Übersicht über die
neuere algebraische Zahlentheorie.
3612 Yuan Wang: Diophantine equations and inequalities in algebraic
number fields. Springer 1991.
2070 Benne de Weger: Algorithms for diophantine equations.
Centrum for Wiskunde 1989. ISBN 90-6196-375-3.
Andrew Wiles: Modular elliptic curves and Fermat's Last Theorem.
Manuscript, october 1994. This long paper "announces a proof of, among
other things, Fermat's Last Theorem, relying on the second one (short)
[Taylor/Wiles, Ring theoretic properties of certain Hecke algebras]
for one crucial step. As most of you know, the argument described by
Wiles in his Cambridge lectures turned out to have a serious gap, namely
the construction of an uler system. After trying unsuccessfully to repair
that construction, Wiles went back to a different approach, which he had
tried earlier but abandoned in favor of the Euler system idea. He was
able to complete his proof, under the hypothesis that certain Hecke
algebras are local complete intersections. This and the rest of the
ideas described in Wiles' Cambridge lectures are written up in the
first manuscript. Jointly, Taylor and Wiles establish the necessary
property of the Hecke algebras in the second paper. The overall outline
of the argument is similar to the one Wiles described in Cambridge.
The new approach turns out to be significantly simpler and shorter
than the original one, because of the removal of the Euler system.
(In fact, after seeing these manuscripts Faltings has apparently come
up with a further significant simplification of that part of the argument.)
Versions of these manuscripts have been in the hands of a small number
of people for (in some cases) a few weeks. While it is wise to be
cautious for a little while longer, there is certainly reason for
optimism." (Karl Rubin).
Andrew Wiles: Modular elliptic curves and Fermat's last theorem.
Ann. Math. 141 (1995), 443-551.
1947 H. Woll: Reductions among number theoretic problems.
Inf. and Computing 72 (1987), 167-179.
7915 Gisbert Wuestholz: Ausgewaehlte Kapitel aus Zahlentheorie und Geometrie.
Internet 1996, 37p.
Zannier: Remarks on a question of Skolem about the integer solutions of
x1x2-x3x4=1. Acta Arithm. 78 (1996), 153-164.