IBM Ponder This #278 · June 2021

The numbers 32993 and 2097593 have the following curious properties:

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

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

For example, for 32993 we have:

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

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

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

32993
2 15
143 112
15 128
25 68

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

Solution

Best opened after a real attempt

To be added.