Tijdeman's theorem
{{short description|There are at most a finite number of consecutive powers}}
In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation
:
for exponents n and m greater than one, is finite.{{ citation | pages=352 | title=Rational Number Theory in the 20th Century: From PNT to FLT | series=Springer Monographs in Mathematics | first=Wladyslaw | last=Narkiewicz | publisher=Springer-Verlag | year=2011 | isbn=978-0-857-29531-6 }}{{citation | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=Springer-Verlag | year=1996 | edition=2nd | isbn=978-3-540-54058-8 | zbl=0754.11020 | page=207 }}
History
The theorem was proven by Dutch number theorist Robert Tijdeman in 1976,{{citation |first=Robert |last=Tijdeman |title=On the equation of Catalan |journal=Acta Arithmetica |volume=29 |issue=2 |year=1976 |pages=197–209 |doi= 10.4064/aa-29-2-197-209| zbl=0286.10013 |doi-access=free }} making use of Baker's method in transcendental number theory to give an effective upper bound for x,y,m,n. Michel Langevin computed a value of exp exp exp exp 730 for the bound.{{citation | title=13 Lectures on Fermat's Last Theorem | first=Paulo | last=Ribenboim | authorlink=Paulo Ribenboim | publisher=Springer-Verlag | year=1979 | isbn=978-0-387-90432-0 | zbl=0456.10006 | page=236 }}{{citation
| last = Langevin | first = Michel | author-link = Michel Langevin (mathematician)
| issue = G12
| journal = Séminaire Delange-Pisot-Poitou, 17e Année (1975/76), Théorie des Nombres
| mr = 0498426
| title = Quelques applications de nouveaux résultats de Van der Poorten
| volume = 2
| year = 1977}}
Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu.{{citation | first=Tauno |last =Metsänkylä | url=http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf | title=Catalan's conjecture: another old Diophantine problem solved | journal=Bulletin of the American Mathematical Society | volume=41 | year=2004 | issue=1 | pages=43–57 | doi=10.1090/S0273-0979-03-00993-5 | doi-access=free }} Mihăilescu's theorem states that there is only one member of the set of consecutive power pairs, namely 9=8+1.{{citation
| last = Mihăilescu | first = Preda | author-link = Preda Mihăilescu
| doi = 10.1515/crll.2004.048
| issue = 572
| journal = Journal für die reine und angewandte Mathematik
| mr = 2076124
| pages = 167–195
| title = Primary Cyclotomic Units and a Proof of Catalan's Conjecture
| volume = 2004
| year = 2004}}
Generalized Tijdeman problem
That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of 1 by any other difference k and ask for the number of solutions
of
:
with n and m greater than one we have an unsolved problem,{{cite book | last1=Shorey | first1=Tarlok N. | last2=Tijdeman | first2=Robert | author2-link=Robert Tijdeman | title=Exponential Diophantine equations | series=Cambridge Tracts in Mathematics | volume=87 | publisher=Cambridge University Press | year=1986 | isbn=978-0-521-26826-4 | zbl=0606.10011 | page=202 |mr=0891406}} called the generalized Tijdeman problem. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture, stating that the equation only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture.{{harvtxt|Narkiewicz|2011}}, pp. 253–254