Difference between revisions of "Zsigmondy's Theorem"
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| − | '''Zsigmondy's Theorem''' states that, for positive [[relatively prime]] integers <math>a</math>, <math>b</math>, and <math>n</math> with <math>a>b</math>, there exists a prime number <math>p</math> (called a ''primitive prime factor'') such that <math>p|(a^n-b^n)</math> but <math>p\ | + | '''Zsigmondy's Theorem''' states that, for positive [[relatively prime]] integers <math>a</math>, <math>b</math>, and <math>n</math> with <math>a>b</math>, there exists a prime number <math>p</math> (called a ''primitive prime factor'') such that <math>p|(a^n-b^n)</math> but <math>p\nmid(a^k-b^k)</math> for all positive integers <math>k<n</math> EXCEPT (i) if <math>n=a-b=1</math>, (ii) if <math>n=2</math> and <math>a+b</math> is a power of <math>2</math>, or (iii) if <math>n=6</math>, <math>a=2</math>, and <math>b=1</math>. This theorem can sometimes be used to prove that no more solutions exist to [[Diophantine equations]]. |
== Example == | == Example == | ||
| − | We desire to find all solutions <math>(x,y)</math> to the Diophantine <math>3^x-1=5^y</math>. We notice that the first integer <math>x</math> for which <math>5|3^x-1</math> is <math>x=4</math>, which produces <math>3^4-1=80\neq 5^y</math>. Now, by Zsigmondy's Theorem (the exceptions do not apply here), for any positive integers <math>n\ | + | We desire to find all solutions <math>(x,y)</math> to the Diophantine <math>3^x-1=5^y</math>. We notice that the first integer <math>x</math> for which <math>5|3^x-1</math> is <math>x=4</math>, which produces <math>3^4-1=80\neq 5^y</math>. Now, by Zsigmondy's Theorem (the exceptions do not apply here), for any positive integers <math>n\ge 4</math>, there must exist a primitive prime factor for each of the terms <math>3^n-1^n.</math> However, the right hand side of the equation only contains prime factors of <math>5</math>, and so there are <math>\boxed{\text{no solutions}}</math> to this Diophantine equation. |
== Problems == | == Problems == | ||
* Find all solutions <math>(x,y,z)</math> to the Diophantine equation <math>7^x-1=2\cdot3^y\cdot5^z</math>. | * Find all solutions <math>(x,y,z)</math> to the Diophantine equation <math>7^x-1=2\cdot3^y\cdot5^z</math>. | ||
| + | * Prove that the equation <math>(x+y)^n=x^m+y^m</math> has a unique solution in integers with <math>x>y>0</math> and <math>m,n>1</math>. | ||
== See Also == | == See Also == | ||
Latest revision as of 03:08, 30 August 2025
Zsigmondy's Theorem states that, for positive relatively prime integers 
, 
, and 
with 
, there exists a prime number 
(called a primitive prime factor) such that 
but 
for all positive integers 
EXCEPT (i) if 
, (ii) if 
and 
is a power of 
, or (iii) if 
, 
, and 
. This theorem can sometimes be used to prove that no more solutions exist to Diophantine equations.
Example
We desire to find all solutions 
to the Diophantine 
. We notice that the first integer 
for which 
is 
, which produces 
. Now, by Zsigmondy's Theorem (the exceptions do not apply here), for any positive integers 
, there must exist a primitive prime factor for each of the terms 
However, the right hand side of the equation only contains prime factors of 
, and so there are 
to this Diophantine equation.
Problems
- Find all solutions
to the Diophantine equation 
. - Prove that the equation
has a unique solution in integers with 
and 
.
to the Diophantine equation 
.
has a unique solution in integers with 
and 
.See Also
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