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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)

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Solution A much faster way | Comment 5 of 8 |
(In reply to re: Corrigendum by Charlie)

After realizing that factoring was slow, and the numbers are so sparse, I realized the way to go was to form powers and see which converted F to an integral C. Instead of taking what would have been days, it was over in less than a second.

count=0; set=double.empty(0,4);
for base=2:round(sqrt(1000000000))
    F=base;
    while F<1000000000
        F=F*base;
        C=(F-32)*5/9;
        if C==round(C)
            count=count+1;
            set=[set; C F base log(F)/log(base) ]
            disp([C F base log(F)/log(base)])
        end
    end
end
set=sortrows(set)


         C             F        base   exponent
            0              32      2     5
         1120            2048      2    11
        43385           78125      5     7
        72800          131072      2    17
        89455          161051     11     5
      1777760         3200000     20     5
      4660320         8388608      2    23
     11395065        20511149     29     5
     44019520        79235168     38     5
     58563040       105413504     14     7
    127413875       229345007     47     5
    298261600       536870912      2    29
    305962080       550731776     56     5 * see note below
    644605885      1160290625     65     5
    678168385      1220703125      5    13
   1232781440      2219006624     74     5
   1891569675      3404825447     23     7
   2188355895      3939040643     83     5
   3661564000      6590815232     92     5
   5838944705     10510100501    101     5
   8947277760     16105100000    110     5
  13257520315     23863536599    119     5
  19088743520     34359738368    128     5
  26812069125     48261724457    137     5
  36854606080     66338290976    146     5
  49703387135     89466096875    155     5
  65909305440    118636749824    164     5
  86091051145    154963892093    173     5

That went a little overboard. The set of values beyond 10^9 may very well be incomplete, lacking some higher powers of lower numbers, as it wasn't designed to go over a billion. So

    305962080       550731776     56     5

is the last in the set expected to be complete.


Edited on January 9, 2022, 8:39 pm
  Posted by Charlie on 2022-01-09 20:31:53

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