In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Goormaghtigh conjecture is a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
named for the
Belgian
Belgian may refer to:
* Something of, or related to, Belgium
* Belgians, people from Belgium or of Belgian descent
* Languages of Belgium, languages spoken in Belgium, such as Dutch, French, and German
*Ancient Belgian language, an extinct languag ...
mathematician
René Goormaghtigh
René Goormaghtigh (13 October 1893, Ostend – 10 February 1960, Ixelles) was a Belgian engineer, after whom the Goormaghtigh Conjecture is named.
Goormaghtigh studied at Ghent University, gaining a Diploma in Civil Engineering from the Central ...
. The conjecture is that the only non-trivial
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
solutions of the
exponential Diophantine equation
:
satisfying
and
are
:
and
:
Partial results
showed that, for each pair of fixed exponents
and
, this equation has only finitely many solutions. But this
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
depends on
Siegel's finiteness theorem, which is ineffective. showed that, if
and
with
,
, and
, then
is bounded by an
effectively computable
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can d ...
constant depending only on
and
. showed that for
and odd
, this equation has no solution
other than the two solutions given above.
Balasubramanian
Balasubramaniam or Balasubramanian ( te, బాల à°¸à±à°¬à±à°°à°¹à±à°®à°£à±à°¯à°‚; ta, பாலசà¯à®ªà¯à®°à®®à®£à®¿à®¯à®®à¯; kn, ಬಾಲಸà³à²¬à³à²°à²¹à³à²®à²£à³à²¯à²‚; ml, ബാലസàµà´¬àµà´°à´¹àµà´®à´£àµà´¯à´‚) is a mal ...
and Shorey proved in 1980 that there are only finitely many possible solutions
to the equations with
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
divisors of
and
lying in a given finite set and that they may be effectively computed.
showed that, for each fixed
and
, this equation has at most one solution.
For fixed ''x'' (or ''y''), equation has at most 15 solutions, and at most two unless ''x'' is either odd
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
times a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
, or in the finite set , in which case there are at most three solutions. Furthermore, there is at most one solution if the odd part of ''n'' is squareful unless ''n'' has at most two distinct odd prime factors or ''n'' is in a finite set .
Application to repunits
The Goormaghtigh conjecture may be expressed as saying that 31 (111 in
base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2) are the only two numbers that are
repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreat ...
s with at least 3 digits in two different bases.
See also
*
Feit–Thompson conjecture
In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by . The conjecture states that there are no distinct prime numbers ''p'' and ''q'' such that
:\frac divides \frac.
If the conjecture were true, it would ...
References
* Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88
*
*
*
*
*
*
*
* {{cite journal , first=Pingzhi , last=Yuan , title=On the diophantine equation
, journal=J. Number Theory , volume=112 , year=2005 , pages=20–25 , doi=10.1016/j.jnt.2004.12.002 , mr=2131139 , doi-access=free
Diophantine equations
Conjectures
Unsolved problems in number theory