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In 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 1 ...
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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 language ...
mathematician René Goormaghtigh. 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 In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
:\frac = \frac satisfying x>y>1 and n,m>2 are :\frac = \frac = 31 and :\frac = \frac = 8191.


Partial results

showed that, for each pair of fixed exponents m and n, this equation has only finitely many solutions. But this proof depends on Siegel's finiteness theorem, which is ineffective. showed that, if m-1=dr and n-1=ds with d \ge 2, r \ge 1, and s \ge 1, then \max(x,y,m,n) 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 r and s. showed that for m=3 and odd n, this equation has no solution (x,y,n) other than the two solutions given above. Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions (x,y,m,n) 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 way ...
divisors of x and y lying in a given finite set and that they may be effectively computed. showed that, for each fixed x and y, 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, ...
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-negati ...
, 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 ''Recrea ...
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 \tfrac{x^3-1}{x-1}=\tfrac{y^n-1}{y-1} , 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