Problems & Puzzles: Puzzles

Puzzle 625. Sum of squares of prime divisors After the generous acceptance of my friend Frank Rubin, I bring -from one of his always interesting pages, Contest Center- the following nice and unsolved puzzle: Which whole numbers are equal to the sum of the squares of their prime divisors? For example: 16=2×2×2×2=2²+2²+2²+2² and 27=3×3×3=3²+3²+3². Q. Are there more cases, or these two are the unique examples?

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Contributions came from J. K. Andersen

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Andersen wrote:

There are no other cases below 10^11.

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Giorgos Kalogeropoulos wrote on Set 18, 2020:

31980599086523546548147351491272676211458715997231784732063781637489066745716387150725397533911738728
841877998280065787261857594370930355385463920685184046649444274178977072736045 =
3 x 3 x 5 x 3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489 x 178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609 =
3^2 + 3^2 + 5^2 +  3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489^2 +  178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609 ^2

 

My method:

Firstly I found some tuples (containing only primes and 1) that fit the equation Sum(squares of factors) = Product(factors)

{1, 1, 3, 3, 5},{1, 1, 1, 2, 2, 2, 3}, {1, 1, 1, 2, 2, 2, 5}....
 

So starting from 1^2+1^2+3^2+3^2+5^2 = 1*1*3*3*5 , I kept {3,3,5} and tried to find 2 other primes (x and y) using Mathematica

The following program tried to solve the equation 3^2+3^2+5^2+x^2+y^2 = 3*3*5*x*y for x,y greater than different powers of 10

It searched the first 100 powers of 10 trying to find any x or any y solutions that were primes (NOT both of them should be primes)   

 

ClearAll[x, y]
Monitor[Union[Table[Select[Flatten[{x, y} /. FindInstance[x^2 + y^2 - 45 x*y + 43 == 0 && x > 10^k && y > 10^k, {x, y},Integers]], PrimeQ], {k, 100}]], k]        
 

this returned three primes   

{{},{1979},{736938099701099},{178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609}}     
 

then, I tried the same thing using these primes as x and see if I could get a y that was a prime    

When I used the big prime as x, 

x=178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609;
y/.FindInstance[x^2+y^2-45*x*y+43==0&&y>10^85,{y},Integers,5]      
 

I got 2 results for y,   

{3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489, 

 8041440874939023184572179930735391848546966482174522004565520846483281791738161037022885916}   
 

and luckily the first one was Prime!        

 

So, the final answer is 3*3*5*x*y      

 

319805990865235465481473514912726762114587159972317847320637816374890667457163871507253975
33911738728841877998280065787261857594370930355385463920685184046649444274178977072736045 = 3 x 3 x 5 x 3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489 x 178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609 = 3^2 + 3^2 + 5^2 +  3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489^2 +  178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609 ^2

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