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 Exponentiation Equals Concatenation (Posted on 2020-11-14)
Solve ab = concatenate(a,b) or prove that there is no solution.

For example if 210 were equal to 210, this would be a solution.

 No Solution Yet Submitted by Larry Rating: 2.0000 (2 votes)

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 Narrowing the domain & range | Comment 1 of 3
No limits are placed on a and b. But (let a&b = a concatonate  b)
1) a,b must be real or they won't concatenate
2) if a<0, a must be an integer, then b must be a non-negative integer, therefore -a, which is >0 in this case, is also a solution, so we can limit our search to look for solutions in the non-negative real numbers for a.
3) a cannot be zero, becasue 0^0 is undefined
4) 0<a<1 is not a solution, becasue b must be a non-negative integer to concatenate with a decimal, and in this case a^b decreases while a&b increases.  b=0 never works
5) a>=1  implies  b must be >=0 or a&b won't exist, a=1 is not a solution
5a) a= integer > 1.  b= integer=0 or 1 are never solutions for any such a
5b) a=integer >1, b= integer>=2.  Easy to see no solutions by inspection
5c) a=integer>1, b=non integer>0  ????
5d) a=rational non-integer >1.  b must be an integer>1 ????
5e) a= irrational number >1,b must be integer >1 ????

SO
- a is a non integer >1, b must be an integer>1 - SUSPECT NO SOLUTION?
OR
a=integer>1, b=positive non-integer ???

 Posted by Kenny M on 2020-11-14 19:48:20

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