Find two 4 digit numbers that multiply to give 4^8 + 6^8 + 9^8

[Genady]

The whole puzzle is in the title. It took a friend of mine, in his own words, "3 pages of factoring (correcting and replacing)". It took me 6 lines. Take a challenge?

[TheVat]

Would finding low prime factors be of any use?  Maybe not.  

[Genady]

9 minutes ago, TheVat said:

Would finding low prime factors be of any use?  Maybe not.  

I don't know how I would do this except by going backward, i.e. after finding the answer to the puzzle looking for its prime factors.

[Genady]

Step 1:

4⁸ + 6⁸ + 9⁸ = 2^2*8 + 2⁸*3⁸ + 3^2*8 

[Genady]

Step 1:           4⁸ + 6⁸ + 9⁸ = 2^2*8 + 2⁸*3⁸ + 3^2*8 

Step 1.5:                             = (2⁸)² + 2⁸*3⁸ + (3⁸)² 

[Genady]

On 1/1/2022 at 8:11 PM, TheVat said:

Would finding low prime factors be of any use?  Maybe not.  

Did you mean these "low prime factors" -- see the previous post? If so, I apologize. My mistake, I didn't understand you. You were right.

From this point, you need only one "trick" more, and not a very uncommon one.

[Genady]

Step 1:           4⁸ + 6⁸ + 9⁸ = 2^2*8 + 2⁸*3⁸ + 3^2*8 

Step 1.5:                             = (2⁸)² + 2⁸*3⁸ + (3⁸)² 

Step 2:                               = (2⁸)² + 2*2⁸*3⁸ + (3⁸)² - 2⁸*3⁸

[Genady]

Step 1:           4⁸ + 6⁸ + 9⁸ = 2^2*8 + 2⁸*3⁸ + 3^2*8 

Step 1.5:                             = (2⁸)² + 2⁸*3⁸ + (3⁸)² 

Step 2:                               = (2⁸)² + 2*2⁸*3⁸ + (3⁸)² - 2⁸*3⁸

Step 2.5:                            = (2⁸ + 3⁸)² - (2⁴*3⁴)²

[Genady]

Step 1:           4⁸ + 6⁸ + 9⁸ = 2^2*8 + 2⁸*3⁸ + 3^2*8 

Step 1.5:                             = (2⁸)² + 2⁸*3⁸ + (3⁸)² 

Step 2:                               = (2⁸)² + 2*2⁸*3⁸ + (3⁸)² - 2⁸*3⁸

Step 2.5:                            = (2⁸ + 3⁸)² - (2⁴*3⁴)²

Step 3:                            = (2⁸ + 3⁸ + 2⁴*3⁴)(2⁸ + 3⁸ - 2⁴*3⁴) = 8113*5521