[ *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 = 23^{2}

184^{2} + 345^{2} = 391^{2}

184^{3} + 345^{3} = 6877^{2}

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 = 73^{2}

889^{2} + 4440^{2} = not square

889^{3} + 4440^{3} = 297037^{2}

2044 + 3285 = 73^{2}

2044^{2} + 3285^{2} = 3869^{2}

2044^{3} + 3285^{3} = not square

2409 + 2920 = 73^{2}

2409^{2} + 2920^{2} = not square

2409^{3} + 2920^{3} = 197173^{2}

3042 + 3042 = 78^{2} {identical !}

3042^{2} + 3042^{2} = not square

3042^{3} + 3042^{3} = 237276^{2}

1729 + 5160 = 83^{2} {powers of Ramanujan's 1729}

1729^{2} + 5160^{2} = not square

1729^{3} + 5160^{3} = 377567^{2}

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:

k^{2} * 23 * 8; k^{2} * 23 * 15, with k = 1, 2, 3...

The proof:

k^{2} * 23 * 8 + k^{2} * 23 * 15 = (23 * k)^{2}

k^{4} * 23^{2} * 8^{2} + k^{4} * 23^{2} * 15^{2} = (23 * k^{2})^{2} * (8^{2} + 15^{2}) = (17 * 23 * k^{2})^{2}

k^{6} * 23^{3} * 8^{3} + k^{6} * 23^{3} * 15^{3} = (23 * k^{2})^{3} * (8^{3} + 15^{3}) =

(23 * k^{2})^{3} * (8 + 15) * (8^{2} + 15^{2} – 8 * 15) = (13 * 23^{2} * k^{3})^{2}

Alexandru Petrescu