[ March 21, 2016 ]
The sum of two numbers, their squares and their cubes
Postings from Pieter Post

The pair (184, 345) possesses the following property:
The sum of the numbers is a square, the sum of their squares is a square
and the sum of their cubes is also a square.

184 + 345 = 23²
184² + 345² = 391²
184³ + 345³ = 6877²

This pair is the only one I could find, at least
with a greatest common divisor of 1. (➜ °)
Naturally (736, 1380) and all quadratic multiples of
(184, 345) fulfil the conditions,
but I could find only one primitive solution.

Maybe someone among my readers has enough computing
capacity to discover a second pair ?

[ April 28, 2016 ]

Delighted to see the presentation made by Mr Pieter Post
regarding the admiring properties of certain pair of numbers
such as 184,345; 736,1380.
My attempts to find second set of such pair of numbers have
not fully materialised so far.
Selected pairs from dozens of my humble findings are enclosed
for your kind perusal.
These were found by manual manipulations on Excel sheet.

889 + 4440 = 73²
889² + 4440² = not square
889³ + 4440³ = 297037²

2044 + 3285 = 73²
2044² + 3285² = 3869²
2044³ + 3285³ = not square

2409 + 2920 = 73²
2409² + 2920² = not square
2409³ + 2920³ = 197173²

3042 + 3042 = 78² {identical !}
3042² + 3042² = not square
3042³ + 3042³ = 237276²

1729 + 5160 = 83² {powers of Ramanujan's 1729}
1729² + 5160² = not square
1729³ + 5160³ = 377567²

I can submit to you these set of pairs on reaching 100^2.

B.S.Rangaswamy

[ May 1, 2016 ]

Patrick,
I only just saw WON Plate 198 today.
Some initial things I notice are

(184 * 2) - 345 = 23

391 / 23 = 17

6877 / 23 = 299

I don't know if these things mean anything or not
but they are curious.

Jeff Heleen

[ June 6, 2016 ]

I wish to share the following general inference
from my studies of the postings by Mr Pieter Post

“ When sum of two identical numbers is a square,
sum of their respective cubes always is a square ”

as in:

50 + 50 = 10^2
50^3 + 50^3 = 500^2

6272 + 6272 = 112^2
6272^3 + 6272^3 = 702464^2

2238728 + 2238728 = 2116^2
2238728^3 + 2238728^3 = 4737148448^2

Generalized:

A^2/2 + A^2/2 = A^2
(A^2/2)^3 + (A^2/2)^3 =
(A^6/8) + (A^6/8) = A^6/4 ➜ Square of (A^3/2)

B.S.Rangaswamy

[ December 14, 2021 ]

(➜ °) The relations are true, but it's a mistake,
gcd(184,345) = 23 and not 1 !
184 = 8 * 23; 345 = 15 * 23

The general solutions are:
k² * 23 * 8; k² * 23 * 15, with k = 1, 2, 3...

The proof:
k² * 23 * 8 + k² * 23 * 15 = (23 * k)²

k⁴ * 23² * 8² + k⁴ * 23² * 15² = (23 * k²)² * (8² + 15²) = (17 * 23 * k²)²

k⁶ * 23³ * 8³ + k⁶ * 23³ * 15³ = (23 * k²)³ * (8³ + 15³) =
(23 * k²)³ * (8 + 15) * (8² + 15² – 8 * 15) = (13 * 23² * k³)²

Alexandru Petrescu