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Perfect Powered Temperature (Posted on 2022-01-09) Difficulty: 3 of 5
Each of M and N is a nonnegative integer such that:

Mo Celsius = No Fahrenheit.

Determine all possible values of N<10 such that N is a perfect power.

*** F = (9/5)*C +32, where F denotes degree Fahrenheit and C denotes degree Celsius.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 1 of 8
Ignoring N<10 as a typo, did up to 1 million.

for fahr=0:1000000
   celsius=(fahr-32)*5/9;
   if celsius==floor(celsius)
      f=factor(fahr); counted=0; prev=0;      
      pwrCt=[];
      for i=1:length(f)
          if f(i)~=prev
             counted=counted+1;
             pwrCt(counted)=1;
             prev=f(i);
          else
              pwrCt(counted)=pwrCt(counted)+1;
          end
      end
      g=gcd(sym(pwrCt));
      if g>1
         base=fahr^(1/g));      
         disp([celsius fahr base g]) 
      end
   end
end

finds the following non-negative values

[0, 32, 2, 5]
[1120, 2048, 2, 11]
[43385, 78125, 5, 7]
[72800, 131072, 2, 17]
[89455, 161051, 11, 5]
 
In each line the three numbers are: Celsius, Fahrenheit, the base of the power, and the perfect power, such as 161051 is 11^5. 

A similar attempt with negative Fahrenheit temperatures fails, as the powers are all even, producing positive rather than negative values:

[-240, -400, 20, 2]
[-160, -256, 2, 8]
[-85, -121, 11, 2]
[-45, -49, 7, 2]
[-20, -4, 2, 2]

  Posted by Charlie on 2022-01-09 10:05:17
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