Ponder This Challenge - June 2021 - Primes that are sums of powers

The numbers 32993 and 2097593 have the following curious properties:

1. They are both prime
2. They are both of the form $\left(a^b+b^a\right)$ for $\left(a,b > 1\right)$ integers
3. They are both of the form $\left(x^2+y^2\right)$ for $\left(x,y\right)$ integers
4. They are both of the form $\left(x^2+2y^2\right)$ for $\left(x,y\right)$ integers
5. They are both of the form $\left(x^2+7y^2\right)$ for $\left(x,y\right)$ integers

Where in equations 3-5 the x,y values can be different from equation to equation.

For example, for 32993 we have:

• $\left(2^{15}+15^2 = 32993\right)$
• $\left(143^2+112^2 = 32993\right)$
• $\left(15^2+2\cdot 128^2 = 32993\right)$
• $\left(25^2+7\cdot 68^2 = 32993\right)$

Your goal: Find another number satisfying those properties. Supply your answer in the following format: first the number itself, then the $\left(a,b\right)$ values that plug into the $\left(a^b+b^a\right)$ equation, and then the $\left(x,y\right)$ values solving equations 3-5.

For example, for 32993, the solution is given as:

A bonus "*" will be given for finding a number greater than googol $(\left(10^{100}\right))$ that satisfies the properties.

Solution

• Click here to view the solution

  While brute force is possible, certainly for the non-asterisked part of the problem, a quick way to determine possible representations as $x^2+ny^2$ is using Cornacchia's algorithm, which can be easily implemented without relying on any number-theory-specific math library.

  A simple solution with more than 100 digits arises for a=34 and b=75, giving the following solution:

  7259701736680389461922586102342375953169154793471358981661239413987142371528493467259545421437269088935158394128249  
  (34, 75)  
  (2341523227685802973317116937233099044272373695299363945845, 1333030648893058553789277061401358577881136005688481248832)  
  (1688720447290015642452516760229506509635032011397287059257, 1484574852876236578030099116366398251085237658488907102090)  
  (1393905404716216131892233482794414472900782174600716102781, 871511778412605044471256876208116162244764418001268613572)
  

Solvers

• *Uoti Urpala (31/5/2021 08:19 PM IDT)
• *Bert Dobbelaere (31/5/2021 10:07 PM IDT)
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