Which two 4-digit numbers multiply to give 4^8 + 6^8 + 9^8?

[Genady]

That is, 4⁸ + 6⁸ + 9⁸.

Edited November 5, 2024 by Genady

[sethoflagos]

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Terms are first three terms of a geometric series a = 4⁸, r = (3/2)⁸, n = 2

Sum = a (rⁿ⁺¹⁾ -1) / (r -1)

         = 4⁸ ((3/2)²⁴ -1) / ((3/2)⁸ -1))

         = 4⁸ ((3²⁴ - 2²⁴) / (3⁸ - 2⁸) * (2⁸ / 2²⁴))

         = (3²⁴ - 2²⁴) / (3⁸ - 2⁸)

         = (3¹⁶ + 3⁸ 2⁸ + 2¹⁶) (3⁸ - 2⁸) / (3⁸ - 2⁸)

         = 3¹⁶ + 3⁸ 2⁸ + 2¹⁶                              ......   which puts me back where I started!!!!!

           = (3⁸ + 2⁸)² - 3⁸ 2⁸                           ....... aha!

           = ((3⁸ + 2⁸) + 3⁴ 2⁴) ((3⁸ + 2⁸) - 3⁴ 2⁴)

           = 8,133 x 5,521

 

Power is failing, so no time to check

[Genady]

  On 11/5/2024 at 5:32 PM, sethoflagos said:

  Reveal hidden contents

Terms are first three terms of a geometric series a = 4⁸, r = (3/2)⁸, n = 2

Sum = a (rⁿ⁺¹⁾ -1) / (r -1)

         = 4⁸ ((3/2)²⁴ -1) / ((3/2)⁸ -1))

         = 4⁸ ((3²⁴ - 2²⁴) / (3⁸ - 2⁸) * (2⁸ / 2²⁴))

         = (3²⁴ - 2²⁴) / (3⁸ - 2⁸)

         = (3¹⁶ + 3⁸ 2⁸ + 2¹⁶) (3⁸ - 2⁸) / (3⁸ - 2⁸)

         = 3¹⁶ + 3⁸ 2⁸ + 2¹⁶                              ......   which puts me back where I started!!!!! Of course.

           = (3⁸ + 2⁸)² - 3⁸ 2⁸                           ....... aha! Yep, complete the square.

           = ((3⁸ + 2⁸) + 3⁴ 2⁴) ((3⁸ + 2⁸) - 3⁴ 2⁴)

           = 8,133 x 5,521 rather 8,113...  Typo perhaps.

 

+1

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[sethoflagos]

  On 11/5/2024 at 6:04 PM, Genady said:

   Typo perhaps

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Indeed. Neither factor can be divisible by 3.

[TheVat]

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Looks like break everything down to powers of 2 and 3, then proceed algebraically.  Would take some time but looks like subtract (2⁴ x 3⁴) from 3⁸+2⁸ and shouid give one factor.  6817 - 1296 = 5521.  Rest is easy.  

 

[Genady]

+1. However, the "take some time" part can be accomplished in three steps, like this:

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2¹⁶+2⁸*3⁸+3¹⁶ = 

2¹⁶+2*2⁸*3⁸+3¹⁶ - 2⁸*3⁸ =

(2⁸+3⁸)² - (2⁴*3⁴)²  =

(2⁸+3⁸ + 2⁴*3⁴)*(2⁸+3⁸ - 2⁴*3⁴)

which gives both factors at once.

Is this what you've meant?