Difference between revisions of "1998 JBMO Problems/Problem 3"
Duck master (talk | contribs) (Created page with "Find all pairs of positive integers <math>(x,y)</math> such that <cmath>x^y = y^{x - y}.</cmath> == Solution == Note that <math>x^y</math> is at least one. Then <math>y^{x -...") |
Duck master (talk | contribs) m (fixed stupid arithmetic mistake. the second solution to the eq is ok now) |
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Since <math>c</math> divides the RHS of this equation, it must divide the LHS. Since <math>\gcd(b, c) = 1</math> by assumption, we must have <math>c = 1</math>, so that the equation reduces to <math>b + 1 = a^b - 1</math>, or <math>b + 2 = a^b</math>. This equation has only the solutions <math>b = 1, a = 3</math> and <math>b = 2, a = 2</math>. | Since <math>c</math> divides the RHS of this equation, it must divide the LHS. Since <math>\gcd(b, c) = 1</math> by assumption, we must have <math>c = 1</math>, so that the equation reduces to <math>b + 1 = a^b - 1</math>, or <math>b + 2 = a^b</math>. This equation has only the solutions <math>b = 1, a = 3</math> and <math>b = 2, a = 2</math>. | ||
− | Therefore, our only solutions are <math>x = 3^{1 + 1} = 9, y = 3^1 = 3</math>, and <math>x = 2^{2+1} = 8, y = 2^2 | + | Therefore, our only solutions are <math>x = 3^{1 + 1} = 9, y = 3^1 = 3</math>, and <math>x = 2^{2+1} = 8, y = 2^1 = 2</math>, and we are done. |
== See also == | == See also == |
Latest revision as of 01:34, 18 September 2020
Find all pairs of positive integers such that
Solution
Note that is at least one. Then is at least one, so .
Write , where . (We know that is nonnegative because .) Then our equation becomes . Taking logarithms base and dividing through by , we obtain .
Since divides the RHS of this equation, it must divide the LHS. Since by assumption, we must have , so that the equation reduces to , or . This equation has only the solutions and .
Therefore, our only solutions are , and , and we are done.
See also
1998 JBMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All JBMO Problems and Solutions |