BrainDen.com - Brain Teasers
• 0

# A rational solution

## Question

For 0<x<y find an integer solution for

x^y = y^x

## 4 answers to this question

• 0

For 0<x<y find an integer solution for

x^y = y^x

##### Share on other sites
• 0
On 10/5/2017 at 10:56 PM, Molly Mae said:

Hide contents

is there another?

##### Share on other sites
• 0

is there another?

Not integers for 0>x>y

Edited by Molly Mae

##### Share on other sites
• 0
Spoiler

Technically, there are an infinite amount of solutions for this equation. To find a solution, you can plug any value for the integer "a" to get a value for x and y.

If "a" is any integer greater than 1, then the the solution is:

x=a-1sqrt(a)

y=a-1sqrt(a^a)

Examples:     a=1.5    x=2.25         y=3.375                       2.25^3.375 = 3.375^2.25 = 15.43888...

a=2       x=2               y=4                              2^4 = 4^2 = 16

a=3       x=sqrt(3)     y=sqrt(27)                  sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3) = 17.36190...

a=4       x=3sqrt(4)   y=3sqrt(256)              3sqrt(4)^3sqrt(256) = 3sqrt(256)^3sqrt(4) = 18.80532...

## Create an account

Register a new account
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://brainden.com/forum/topic/18021-a-rational-solution/?sortby=date", "fetch_time": 1516468121000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-05/segments/1516084889677.76/warc/CC-MAIN-20180120162254-20180120182254-00481.warc.gz", "warc_record_offset": 51322782, "warc_record_length": 15127, "token_count": 410, "char_count": 1150, "score": 3.703125, "int_score": 4, "crawl": "CC-MAIN-2018-05", "snapshot_type": "longest", "language": "en", "language_score": 0.4568580985069275, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00009-of-00064.parquet"}