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interests / rec.puzzles / Re: Mathematical Recreation Problem from 1749

SubjectAuthor
* Mathematical Recreation Problem from 1749leflynn
+* Re: Mathematical Recreation Problem from 1749Mark Brader
|+- Re: Mathematical Recreation Problem from 1749henh...@gmail.com
|`* Re: Mathematical Recreation Problem from 1749leflynn
| `* Re: Mathematical Recreation Problem from 1749Richard Heathfield
|  `* Re: Mathematical Recreation Problem from 1749Ilan Mayer
|   `* Re: Mathematical Recreation Problem from 1749leflynn
|    +* Re: Mathematical Recreation Problem from 1749Richard Heathfield
|    |`* Re: Mathematical Recreation Problem from 1749Richard Tobin
|    | +* Re: Mathematical Recreation Problem from 1749Richard Heathfield
|    | |`* Re: Mathematical Recreation Problem from 1749Mark Brader
|    | | `* Re: Mathematical Recreation Problem from 1749Richard Heathfield
|    | |  `- Re: Mathematical Recreation Problem from 1749Mark Brader
|    | `- Re: Mathematical Recreation Problem from 1749leflynn
|    `* Re: Mathematical Recreation Problem from 1749Ilan Mayer
|     `- Re: Mathematical Recreation Problem from 1749Pancho
`* Re: Mathematical Recreation Problem from 1749Richard Tobin
 +- Re: Mathematical Recreation Problem from 1749Richard Heathfield
 `* Re: Mathematical Recreation Problem from 1749leflynn
  `- Re: Mathematical Recreation Problem from 1749Richard Heathfield

1
Mathematical Recreation Problem from 1749

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Subject: Mathematical Recreation Problem from 1749
From: lefl...@hotmail.com (leflynn)
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 by: leflynn - Fri, 17 Jun 2022 17:54 UTC

To find three such numbers, that the sum or difference of any two of them shall be a square number. I'll post the solution from 1750 in a week.

L. Flynn

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
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 by: Mark Brader - Fri, 17 Jun 2022 21:17 UTC

L. Flynn:
> To find three such numbers, that the sum or difference of any two of
> them shall be a square number.

That's an ambiguous "or". Does it mean that any number which is the
sum or difference of two of the three numbers shall be a square, or does
it mean that it shall be true that either the sum or the difference
is a square?

In other words, if X and Y are two of the numbers and X > Y, is it
required that X+Y and X-Y are both squares, or only one of them?
--
Mark Brader "A clarification is not to make oneself clear.
Toronto It is to PUT oneself IN the clear."
msb@vex.net -- Lynn & Jay, "Yes, Prime Minister"

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sat, 18 Jun 2022 15:33 UTC

On Friday, June 17, 2022 at 2:17:35 PM UTC-7, Mark Brader wrote:
> L. Flynn:
> > To find three such numbers, that the sum or difference of any two of
> > them shall be a square number.

> That's an ambiguous "or". Does it mean that any number which is the
> sum or difference of two of the three numbers shall be a square, or does
> it mean that it shall be true that either the sum or the difference
> is a square?
>
> In other words, if X and Y are two of the numbers and X > Y, is it
> required that X+Y and X-Y are both squares, or only one of them?
> --

https://math.stackexchange.com/questions/1386029/are-there-infinitely-many-pythagorean-triples

( apparently, the question (problem) is asking for something totally different )

so apparently ... this OR means AND

i wonder if this difference in conventions (usage of OR and AND)
is a common and prevalent one. ------- not just in math, but all over

today in math books (and in math classrooms), we rarely encounter this type of confusion / ambiguity

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: lefl...@hotmail.com (leflynn)
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 by: leflynn - Sat, 18 Jun 2022 15:34 UTC

On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
> L. Flynn:
> > To find three such numbers, that the sum or difference of any two of
> > them shall be a square number.
> That's an ambiguous "or". Does it mean that any number which is the
> sum or difference of two of the three numbers shall be a square, or does
> it mean that it shall be true that either the sum or the difference
> is a square?
>
> In other words, if X and Y are two of the numbers and X > Y, is it
> required that X+Y and X-Y are both squares, or only one of them?
> --
X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
L. Flynn

Re: Mathematical Recreation Problem from 1749

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From: rjh...@cpax.org.uk (Richard Heathfield)
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Subject: Re: Mathematical Recreation Problem from 1749
Date: Sat, 18 Jun 2022 17:33:41 +0100
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 by: Richard Heathfield - Sat, 18 Jun 2022 16:33 UTC

On 18/06/2022 4:34 pm, leflynn wrote:
> On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
>> L. Flynn:
>>> To find three such numbers, that the sum or difference of any two of
>>> them shall be a square number.
>> That's an ambiguous "or". Does it mean that any number which is the
>> sum or difference of two of the three numbers shall be a square, or does
>> it mean that it shall be true that either the sum or the difference
>> is a square?
>>
>> In other words, if X and Y are two of the numbers and X > Y, is it
>> required that X+Y and X-Y are both squares, or only one of them?
>> --
> X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.

For all X, Y, and Z in the range 0-1000 and none equal to the
other two, I found no solutions.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

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From: rich...@cogsci.ed.ac.uk (Richard Tobin)
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Subject: Re: Mathematical Recreation Problem from 1749
Date: Sat, 18 Jun 2022 18:52:50 +0000 (UTC)
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 by: Richard Tobin - Sat, 18 Jun 2022 18:52 UTC

In article <0361098f-6745-426a-bf20-e86a100afaa2n@googlegroups.com>,
leflynn <leflynn@hotmail.com> wrote:

>To find three such numbers, that the sum or difference of any two of
>them shall be a square number. I'll post the solution from 1750 in a
>week.

Easier problem: guess who solved it in 1750.

-- Richard

Re: Mathematical Recreation Problem from 1749

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From: rjh...@cpax.org.uk (Richard Heathfield)
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Subject: Re: Mathematical Recreation Problem from 1749
Date: Sat, 18 Jun 2022 21:36:46 +0100
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 by: Richard Heathfield - Sat, 18 Jun 2022 20:36 UTC

On 18/06/2022 7:52 pm, Richard Tobin wrote:
> In article <0361098f-6745-426a-bf20-e86a100afaa2n@googlegroups.com>,
> leflynn <leflynn@hotmail.com> wrote:
>
>> To find three such numbers, that the sum or difference of any two of
>> them shall be a square number. I'll post the solution from 1750 in a
>> week.
>
> Easier problem: guess who solved it in 1750.

Euler, I presume.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: lefl...@hotmail.com (leflynn)
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 by: leflynn - Sun, 19 Jun 2022 02:40 UTC

On Saturday, June 18, 2022 at 2:55:02 PM UTC-4, Richard Tobin wrote:
> In article <0361098f-6745-426a...@googlegroups.com>,
> leflynn <lef...@hotmail.com> wrote:
>
> >To find three such numbers, that the sum or difference of any two of
> >them shall be a square number. I'll post the solution from 1750 in a
> >week.
> Easier problem: guess who solved it in 1750.
>
> -- Richard
Unless one had an encyclopedic knowledge of 18th English speaking persons as a pool (or some set even more limited containing the name of the solver), I would not think guessing the name of the solver I am aware of would be easier than guessing the six square roots. Since, if one guessed three of them correctly, say sqrt(X-Y), sqrt(X-Z) and sqrt(Y-Z) (for X>Y>Z), then the other three could be computed.
L. Flynn

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: ilan_no_...@hotmail.com (Ilan Mayer)
Injection-Date: Sun, 19 Jun 2022 03:55:07 +0000
Content-Type: text/plain; charset="UTF-8"
 by: Ilan Mayer - Sun, 19 Jun 2022 03:55 UTC

On Saturday, June 18, 2022 at 12:33:44 PM UTC-4, Richard Heathfield wrote:
> On 18/06/2022 4:34 pm, leflynn wrote:
> > On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
> >> L. Flynn:
> >>> To find three such numbers, that the sum or difference of any two of
> >>> them shall be a square number.
> >> That's an ambiguous "or". Does it mean that any number which is the
> >> sum or difference of two of the three numbers shall be a square, or does
> >> it mean that it shall be true that either the sum or the difference
> >> is a square?
> >>
> >> In other words, if X and Y are two of the numbers and X > Y, is it
> >> required that X+Y and X-Y are both squares, or only one of them?
> >> --
> > X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
> For all X, Y, and Z in the range 0-1000 and none equal to the
> other two, I found no solutions.
>
> --
> Richard Heathfield
> Email: rjh at cpax dot org dot uk
> "Usenet is a strange place" - dmr 29 July 1999
> Sig line 4 vacant - apply within

[Program as spoiler space]

using System;
using System.Collections.Generic;

namespace SquaresPuzzle
{ class SquaresPuzzle
{
private const int N = 1000;

static void Main()
{
List<int[]> numbers = new List<int[]>();

for (int n1 = 1; n1 <= N; n1++)
{
for (int n2 = 1; n2 < n1; n2++)
{
if (n1 % 2 == n2 % 2)
{
int n1Squared = n1 * n1;
int n2Squared = n2 * n2;
int m1 = (n1Squared + n2Squared) / 2;
int m2 = (n1Squared - n2Squared) / 2;
numbers.Add(new[] { m1, m2 });
}
}
}

Comparer comparer = new Comparer();
numbers.Sort(comparer);
for (int n = 1; n < numbers.Count; n++)
{
if (numbers[n][0] == numbers[n - 1][0])
{
int n1 = numbers[n - 1][1];
int n2 = numbers[n][1];
int[] number1 = new[] { n1, n2 };
int[] number2 = new[] { n2, n1 };
if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
{
Console.WriteLine("{0}, {1}, {2}", numbers[n][0], n1, n2);
}
}
}
}

public class Comparer : IComparer<int[]>
{
public int Compare(int[] x, int[] y)
{
int result = x[0].CompareTo(y[0]);
if (result == 0)
{
result = x[1].CompareTo(y[1]);
}

return result;
}
}
}
}

The program comes up with:
434657, 150568, 420968

Please reply to ilanlmayer at gmail dot com

__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||

Re: Mathematical Recreation Problem from 1749

<t8mk2a$fc8$1@dont-email.me>

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From: rjh...@cpax.org.uk (Richard Heathfield)
Newsgroups: rec.puzzles
Subject: Re: Mathematical Recreation Problem from 1749
Date: Sun, 19 Jun 2022 08:43:05 +0100
Organization: Fix this later
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 by: Richard Heathfield - Sun, 19 Jun 2022 07:43 UTC

On 19/06/2022 3:40 am, leflynn wrote:
> On Saturday, June 18, 2022 at 2:55:02 PM UTC-4, Richard Tobin wrote:
>> In article <0361098f-6745-426a...@googlegroups.com>,
>> leflynn <lef...@hotmail.com> wrote:
>>
>>> To find three such numbers, that the sum or difference of any two of
>>> them shall be a square number. I'll post the solution from 1750 in a
>>> week.
>> Easier problem: guess who solved it in 1750.
>>
>> -- Richard
> Unless one had an encyclopedic knowledge of 18th English speaking persons as a pool (or some set even more limited containing the name of the solver),

....which leaves us with Lambert and Euler (Newton died in 1727
NS). Of the two, Euler is an easy odds-on favourite.

> I would not think guessing the name of the solver I am aware of would be easier than guessing the six square roots.

Oh, but it is, it is.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

<c8c45b51-74c3-45b2-9e66-161c95edd5f3n@googlegroups.com>

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Subject: Re: Mathematical Recreation Problem from 1749
From: lefl...@hotmail.com (leflynn)
Injection-Date: Sun, 19 Jun 2022 11:57:21 +0000
Content-Type: text/plain; charset="UTF-8"
 by: leflynn - Sun, 19 Jun 2022 11:57 UTC

On Saturday, June 18, 2022 at 11:55:09 PM UTC-4, Ilan Mayer wrote:
> On Saturday, June 18, 2022 at 12:33:44 PM UTC-4, Richard Heathfield wrote:
> > On 18/06/2022 4:34 pm, leflynn wrote:
> > > On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
> > >> L. Flynn:
> > >>> To find three such numbers, that the sum or difference of any two of
> > >>> them shall be a square number.
> > >> That's an ambiguous "or". Does it mean that any number which is the
> > >> sum or difference of two of the three numbers shall be a square, or does
> > >> it mean that it shall be true that either the sum or the difference
> > >> is a square?
> > >>
> > >> In other words, if X and Y are two of the numbers and X > Y, is it
> > >> required that X+Y and X-Y are both squares, or only one of them?
> > >> --
> > > X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
> > For all X, Y, and Z in the range 0-1000 and none equal to the
> > other two, I found no solutions.
> >
> > --
> > Richard Heathfield
> > Email: rjh at cpax dot org dot uk
> > "Usenet is a strange place" - dmr 29 July 1999
> > Sig line 4 vacant - apply within
> [Program as spoiler space]
>
> using System;
> using System.Collections.Generic;
>
> namespace SquaresPuzzle
> {
> class SquaresPuzzle
> {
> private const int N = 1000;
>
> static void Main()
> {
> List<int[]> numbers = new List<int[]>();
>
> for (int n1 = 1; n1 <= N; n1++)
> {
> for (int n2 = 1; n2 < n1; n2++)
> {
> if (n1 % 2 == n2 % 2)
> {
> int n1Squared = n1 * n1;
> int n2Squared = n2 * n2;
> int m1 = (n1Squared + n2Squared) / 2;
> int m2 = (n1Squared - n2Squared) / 2;
> numbers.Add(new[] { m1, m2 });
> }
> }
> }
>
> Comparer comparer = new Comparer();
> numbers.Sort(comparer);
> for (int n = 1; n < numbers.Count; n++)
> {
> if (numbers[n][0] == numbers[n - 1][0])
> {
> int n1 = numbers[n - 1][1];
> int n2 = numbers[n][1];
> int[] number1 = new[] { n1, n2 };
> int[] number2 = new[] { n2, n1 };
> if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
> {
> Console.WriteLine("{0}, {1}, {2}", numbers[n][0], n1, n2);
> }
> }
> }
> }
>
> public class Comparer : IComparer<int[]>
> {
> public int Compare(int[] x, int[] y)
> {
> int result = x[0].CompareTo(y[0]);
> if (result == 0)
> {
> result = x[1].CompareTo(y[1]);
> }
>
> return result;
> }
> }
> }
> }
>
> The program comes up with:
> 434657, 150568, 420968
>
> Please reply to ilanlmayer at gmail dot com
>
> __/\__
> \ /
> __/\\ //\__ Ilan Mayer
> \ /
> /__ __\ Toronto, Canada
> /__ __\
> ||

Nice program. The solution found in 1750 was

Answered by Mr. C. Bumpkin . By a method of substitution , too tedious to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers , answering the conditions of the question .

I did not realize that the name was an alias until prodded by Richard Tobin.

L. Flynn

Re: Mathematical Recreation Problem from 1749

<t8n5da$2uq$1@dont-email.me>

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From: rjh...@cpax.org.uk (Richard Heathfield)
Newsgroups: rec.puzzles
Subject: Re: Mathematical Recreation Problem from 1749
Date: Sun, 19 Jun 2022 13:39:05 +0100
Organization: Fix this later
Lines: 115
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 by: Richard Heathfield - Sun, 19 Jun 2022 12:39 UTC

On 19/06/2022 12:57 pm, leflynn wrote:
> On Saturday, June 18, 2022 at 11:55:09 PM UTC-4, Ilan Mayer wrote:
>> On Saturday, June 18, 2022 at 12:33:44 PM UTC-4, Richard Heathfield wrote:
>>> On 18/06/2022 4:34 pm, leflynn wrote:
>>>> On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
>>>>> L. Flynn:
>>>>>> To find three such numbers, that the sum or difference of any two of
>>>>>> them shall be a square number.
>>>>> That's an ambiguous "or". Does it mean that any number which is the
>>>>> sum or difference of two of the three numbers shall be a square, or does
>>>>> it mean that it shall be true that either the sum or the difference
>>>>> is a square?
>>>>>
>>>>> In other words, if X and Y are two of the numbers and X > Y, is it
>>>>> required that X+Y and X-Y are both squares, or only one of them?
>>>>> --
>>>> X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
>>> For all X, Y, and Z in the range 0-1000 and none equal to the
>>> other two, I found no solutions.
>>>
>>> --
>>> Richard Heathfield
>>> Email: rjh at cpax dot org dot uk
>>> "Usenet is a strange place" - dmr 29 July 1999
>>> Sig line 4 vacant - apply within
>> [Program as spoiler space]
>>
>> using System;
>> using System.Collections.Generic;
>>
>> namespace SquaresPuzzle
>> {
>> class SquaresPuzzle
>> {
>> private const int N = 1000;
>>
>> static void Main()
>> {
>> List<int[]> numbers = new List<int[]>();
>>
>> for (int n1 = 1; n1 <= N; n1++)
>> {
>> for (int n2 = 1; n2 < n1; n2++)
>> {
>> if (n1 % 2 == n2 % 2)
>> {
>> int n1Squared = n1 * n1;
>> int n2Squared = n2 * n2;
>> int m1 = (n1Squared + n2Squared) / 2;
>> int m2 = (n1Squared - n2Squared) / 2;
>> numbers.Add(new[] { m1, m2 });
>> }
>> }
>> }
>>
>> Comparer comparer = new Comparer();
>> numbers.Sort(comparer);
>> for (int n = 1; n < numbers.Count; n++)
>> {
>> if (numbers[n][0] == numbers[n - 1][0])
>> {
>> int n1 = numbers[n - 1][1];
>> int n2 = numbers[n][1];
>> int[] number1 = new[] { n1, n2 };
>> int[] number2 = new[] { n2, n1 };
>> if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
>> {
>> Console.WriteLine("{0}, {1}, {2}", numbers[n][0], n1, n2);
>> }
>> }
>> }
>> }
>>
>> public class Comparer : IComparer<int[]>
>> {
>> public int Compare(int[] x, int[] y)
>> {
>> int result = x[0].CompareTo(y[0]);
>> if (result == 0)
>> {
>> result = x[1].CompareTo(y[1]);
>> }
>>
>> return result;
>> }
>> }
>> }
>> }
>>
>> The program comes up with:
>> 434657, 150568, 420968
>>
>> Please reply to ilanlmayer at gmail dot com
>>
>> __/\__
>> \ /
>> __/\\ //\__ Ilan Mayer
>> \ /
>> /__ __\ Toronto, Canada
>> /__ __\
>> ||
>
> Nice program. The solution found in 1750 was
>
> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers , answering the conditions of the question .
>
> I did not realize that the name was an alias until prodded by Richard Tobin.

Alias of whom?

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

<t8n8mb$2p11$1@macpro.inf.ed.ac.uk>

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From: rich...@cogsci.ed.ac.uk (Richard Tobin)
Newsgroups: rec.puzzles
Subject: Re: Mathematical Recreation Problem from 1749
Date: Sun, 19 Jun 2022 13:35:07 +0000 (UTC)
Organization: Language Technology Group, University of Edinburgh
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 by: Richard Tobin - Sun, 19 Jun 2022 13:35 UTC

In article <t8n5da$2uq$1@dont-email.me>,
Richard Heathfield <rjh@cpax.org.uk> wrote:

>> Nice program. The solution found in 1750 was
>>
>> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious
>to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers ,
>answering the conditions of the question .
>>
>> I did not realize that the name was an alias until prodded by Richard Tobin.

>Alias of whom?

It's widely quoted that Frederick the Great considered Euler a
"country bumpkin" - presumably in German - but I have no information
on whether he was in the habit of answering mathematical questions in
the Ladies' Diary:

https://books.google.co.uk/books?id=PLQDyr3T6NQC&pg=PA19

The solution above had been discovered by Ozanam in 1691.

As far as I can see from the interwebs, Euler seems to have been the
first to give the minimum solution and a generating formula.

-- Richard

Re: Mathematical Recreation Problem from 1749

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From: rjh...@cpax.org.uk (Richard Heathfield)
Newsgroups: rec.puzzles
Subject: Re: Mathematical Recreation Problem from 1749
Date: Sun, 19 Jun 2022 15:01:21 +0100
Organization: Fix this later
Lines: 31
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 by: Richard Heathfield - Sun, 19 Jun 2022 14:01 UTC

On 19/06/2022 2:35 pm, Richard Tobin wrote:
> In article <t8n5da$2uq$1@dont-email.me>,
> Richard Heathfield <rjh@cpax.org.uk> wrote:
>
>>> Nice program. The solution found in 1750 was
>>>
>>> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious
>> to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers ,
>> answering the conditions of the question .
>>>
>>> I did not realize that the name was an alias until prodded by Richard Tobin.
>
>> Alias of whom?
>
> It's widely quoted that Frederick the Great considered Euler a
> "country bumpkin"

Ha! Thank you; I didn't know that.

So I was licking my wounds in error!

> As far as I can see from the interwebs, Euler seems to have been the
> first to give the minimum solution and a generating formula.

In 1750, who else could it have been?

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: lefl...@hotmail.com (leflynn)
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 by: leflynn - Sun, 19 Jun 2022 14:18 UTC

On Sunday, June 19, 2022 at 9:40:03 AM UTC-4, Richard Tobin wrote:
> In article <t8n5da$2uq$1...@dont-email.me>,
> Richard Heathfield <r...@cpax.org.uk> wrote:
>
> >> Nice program. The solution found in 1750 was
> >>
> >> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious
> >to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers ,
> >answering the conditions of the question .
> >>
> >> I did not realize that the name was an alias until prodded by Richard Tobin.
>
> >Alias of whom?
> It's widely quoted that Frederick the Great considered Euler a
> "country bumpkin" - presumably in German - but I have no information
> on whether he was in the habit of answering mathematical questions in
> the Ladies' Diary:
>
> https://books.google.co.uk/books?id=PLQDyr3T6NQC&pg=PA19
>
> The solution above had been discovered by Ozanam in 1691.
>
> As far as I can see from the interwebs, Euler seems to have been the
> first to give the minimum solution and a generating formula.
>
> -- Richard
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.352.5649&rep=rep1&type=pdf

In particular, Landen used the pseudonyms Sir Stately Stiff, Peter Walton, Waltoniensis, C. Bumpkin, and Peter Puzzlem, who, collectively, proposed ten problems and answered seventeen.

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
From: ilan_no_...@hotmail.com (Ilan Mayer)
Injection-Date: Sun, 19 Jun 2022 14:33:50 +0000
Content-Type: text/plain; charset="UTF-8"
 by: Ilan Mayer - Sun, 19 Jun 2022 14:33 UTC

On Sunday, June 19, 2022 at 7:57:23 AM UTC-4, leflynn wrote:
> On Saturday, June 18, 2022 at 11:55:09 PM UTC-4, Ilan Mayer wrote:
> > On Saturday, June 18, 2022 at 12:33:44 PM UTC-4, Richard Heathfield wrote:
> > > On 18/06/2022 4:34 pm, leflynn wrote:
> > > > On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
> > > >> L. Flynn:
> > > >>> To find three such numbers, that the sum or difference of any two of
> > > >>> them shall be a square number.
> > > >> That's an ambiguous "or". Does it mean that any number which is the
> > > >> sum or difference of two of the three numbers shall be a square, or does
> > > >> it mean that it shall be true that either the sum or the difference
> > > >> is a square?
> > > >>
> > > >> In other words, if X and Y are two of the numbers and X > Y, is it
> > > >> required that X+Y and X-Y are both squares, or only one of them?
> > > >> --
> > > > X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
> > > For all X, Y, and Z in the range 0-1000 and none equal to the
> > > other two, I found no solutions.
> > >
> > > --
> > > Richard Heathfield
> > > Email: rjh at cpax dot org dot uk
> > > "Usenet is a strange place" - dmr 29 July 1999
> > > Sig line 4 vacant - apply within
> > [Program as spoiler space]
> >
> > using System;
> > using System.Collections.Generic;
> >
> > namespace SquaresPuzzle
> > {
> > class SquaresPuzzle
> > {
> > private const int N = 1000;
> >
> > static void Main()
> > {
> > List<int[]> numbers = new List<int[]>();
> >
> > for (int n1 = 1; n1 <= N; n1++)
> > {
> > for (int n2 = 1; n2 < n1; n2++)
> > {
> > if (n1 % 2 == n2 % 2)
> > {
> > int n1Squared = n1 * n1;
> > int n2Squared = n2 * n2;
> > int m1 = (n1Squared + n2Squared) / 2;
> > int m2 = (n1Squared - n2Squared) / 2;
> > numbers.Add(new[] { m1, m2 });
> > }
> > }
> > }
> >
> > Comparer comparer = new Comparer();
> > numbers.Sort(comparer);
> > for (int n = 1; n < numbers.Count; n++)
> > {
> > if (numbers[n][0] == numbers[n - 1][0])
> > {
> > int n1 = numbers[n - 1][1];
> > int n2 = numbers[n][1];
> > int[] number1 = new[] { n1, n2 };
> > int[] number2 = new[] { n2, n1 };
> > if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
> > {
> > Console.WriteLine("{0}, {1}, {2}", numbers[n][0], n1, n2);
> > }
> > }
> > }
> > }
> >
> > public class Comparer : IComparer<int[]>
> > {
> > public int Compare(int[] x, int[] y)
> > {
> > int result = x[0].CompareTo(y[0]);
> > if (result == 0)
> > {
> > result = x[1].CompareTo(y[1]);
> > }
> >
> > return result;
> > }
> > }
> > }
> > }
> >
> > The program comes up with:
> > 434657, 150568, 420968
> >
> > Please reply to ilanlmayer at gmail dot com
> >
> > __/\__
> > \ /
> > __/\\ //\__ Ilan Mayer
> > \ /
> > /__ __\ Toronto, Canada
> > /__ __\
> > ||
> Nice program. The solution found in 1750 was
>
> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers , answering the conditions of the question .
>
> I did not realize that the name was an alias until prodded by Richard Tobin.
>
> L. Flynn

Revised program:

using System;
using System.Collections.Generic;

namespace SquaresPuzzle
{ class SquaresPuzzle
{
private const int N = 3000;

static void Main()
{
List<int[]> numbers = new List<int[]>();

for (int n1 = 1; n1 <= N; n1++)
{
for (int n2 = 1; n2 < n1; n2++)
{
if (n1 % 2 == n2 % 2)
{
int n1Squared = n1 * n1;
int n2Squared = n2 * n2;
int m1 = (n1Squared + n2Squared) / 2;
int m2 = (n1Squared - n2Squared) / 2;
numbers.Add(new[] { m1, m2 });
}
}
}

Comparer comparer = new Comparer();
numbers.Sort(comparer);
for (int n = 0; n < numbers.Count; n++)
{
int m = n;
while (++m < numbers.Count && numbers[m][0] == numbers[n][0])
{
int n1 = numbers[n][1];
int n2 = numbers[m][1];
int[] number1 = new[] { n1, n2 };
int[] number2 = new[] { n2, n1 };
if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
{
int n0 = numbers[n][0];
int divisor1 = FirstDivisor(n0);
int divisor2 = FirstDivisor(n1);
int divisor3 = FirstDivisor(n2);
if (divisor1 != divisor2 || divisor1 != divisor3)
{
Console.WriteLine("{0}, {1}, {2}", n0, n1, n2);
}
}
}
}
}

private static int FirstDivisor(int number)
{
if (number % 2 == 0)
{
return 2;
}

int root = (int) Math.Sqrt(number);
for (int n = 3; n <= root; n += 2)
{
if (number % n == 0)
{
return n;
}
}

return number;
}

public class Comparer : IComparer<int[]>
{
public int Compare(int[] x, int[] y)
{
int result = x[0].CompareTo(y[0]);
if (result == 0)
{
result = x[1].CompareTo(y[1]);
}

return result;
}
}
}
}

It produces 5 solutions (excluding solutions where all 3 numbers share a divisor), including the original one:
434657, 150568, 420968
733025, 418304, 488000
2399057, 1873432, 2288168
3911913, 1355112, 3788712
5320193, 589568, 1782032

Please reply to ilanlmayer at gmail dot com

__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||

Re: Mathematical Recreation Problem from 1749

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 by: Mark Brader - Mon, 20 Jun 2022 02:27 UTC

Richard Tobin:
>> As far as I can see from the interwebs, Euler seems to have been the
>> first to give the minimum solution and a generating formula.
Richard Heathfield:
> In 1750, who else could it have been?

Well, Gauss was alive then. Elderly, though.
--
Mark Brader, Toronto "Argh! Hoist by my own canard :-) !"
msb@vex.net -- Steve Summit

Re: Mathematical Recreation Problem from 1749

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From: rjh...@cpax.org.uk (Richard Heathfield)
Newsgroups: rec.puzzles
Subject: Re: Mathematical Recreation Problem from 1749
Date: Mon, 20 Jun 2022 07:56:24 +0100
Organization: Fix this later
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 by: Richard Heathfield - Mon, 20 Jun 2022 06:56 UTC

On 20/06/2022 3:27 am, Mark Brader wrote:
> Richard Tobin:
>>> As far as I can see from the interwebs, Euler seems to have been the
>>> first to give the minimum solution and a generating formula.
>
> Richard Heathfield:
>> In 1750, who else could it have been?
>
> Well, Gauss was alive then. Elderly, though.

/The/ Gauss was born in 1777, making him (in 1750) more sort of
youngerly.

Wikipedia lists five other Gausses, none of whom was born earlier
than 1829, making them in 1750 even more youngerly.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

Re: Mathematical Recreation Problem from 1749

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 by: Mark Brader - Mon, 20 Jun 2022 08:47 UTC

Mark Brader:
> > Well, Gauss was alive then. Elderly, though.
Richard Heathfield:
> /The/ Gauss was born in 1777, making him (in 1750) more sort of
> youngerly.

Dang. I looked up his dates, then misread them.
--
Mark Brader | "...not one accident in a hundred deserves the name.
Toronto | [This occurrence] was simply the legitimate result
msb@vex.net | of carelessness." -- Washington Roebling

Re: Mathematical Recreation Problem from 1749

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Subject: Re: Mathematical Recreation Problem from 1749
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 by: Pancho - Mon, 20 Jun 2022 12:32 UTC

On 19/06/2022 15:33, Ilan Mayer wrote:
> On Sunday, June 19, 2022 at 7:57:23 AM UTC-4, leflynn wrote:
>> On Saturday, June 18, 2022 at 11:55:09 PM UTC-4, Ilan Mayer wrote:
>>> On Saturday, June 18, 2022 at 12:33:44 PM UTC-4, Richard Heathfield wrote:
>>>> On 18/06/2022 4:34 pm, leflynn wrote:
>>>>> On Friday, June 17, 2022 at 5:17:35 PM UTC-4, Mark Brader wrote:
>>>>>> L. Flynn:
>>>>>>> To find three such numbers, that the sum or difference of any two of
>>>>>>> them shall be a square number.
>>>>>> That's an ambiguous "or". Does it mean that any number which is the
>>>>>> sum or difference of two of the three numbers shall be a square, or does
>>>>>> it mean that it shall be true that either the sum or the difference
>>>>>> is a square?
>>>>>>
>>>>>> In other words, if X and Y are two of the numbers and X > Y, is it
>>>>>> required that X+Y and X-Y are both squares, or only one of them?
>>>>>> --
>>>>> X+Y, |X-Y|, X+Z, |X-Z|, Y+Z, |Y-Z| are all squares.
>>>> For all X, Y, and Z in the range 0-1000 and none equal to the
>>>> other two, I found no solutions.
>>>>
>>>> --
>>>> Richard Heathfield
>>>> Email: rjh at cpax dot org dot uk
>>>> "Usenet is a strange place" - dmr 29 July 1999
>>>> Sig line 4 vacant - apply within
>>> [Program as spoiler space]
>>>
>>> using System;
>>> using System.Collections.Generic;
>>>
>>> namespace SquaresPuzzle
>>> {
>>> class SquaresPuzzle
>>> {
>>> private const int N = 1000;
>>>
>>> static void Main()
>>> {
>>> List<int[]> numbers = new List<int[]>();
>>>
>>> for (int n1 = 1; n1 <= N; n1++)
>>> {
>>> for (int n2 = 1; n2 < n1; n2++)
>>> {
>>> if (n1 % 2 == n2 % 2)
>>> {
>>> int n1Squared = n1 * n1;
>>> int n2Squared = n2 * n2;
>>> int m1 = (n1Squared + n2Squared) / 2;
>>> int m2 = (n1Squared - n2Squared) / 2;
>>> numbers.Add(new[] { m1, m2 });
>>> }
>>> }
>>> }
>>>
>>> Comparer comparer = new Comparer();
>>> numbers.Sort(comparer);
>>> for (int n = 1; n < numbers.Count; n++)
>>> {
>>> if (numbers[n][0] == numbers[n - 1][0])
>>> {
>>> int n1 = numbers[n - 1][1];
>>> int n2 = numbers[n][1];
>>> int[] number1 = new[] { n1, n2 };
>>> int[] number2 = new[] { n2, n1 };
>>> if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
>>> {
>>> Console.WriteLine("{0}, {1}, {2}", numbers[n][0], n1, n2);
>>> }
>>> }
>>> }
>>> }
>>>
>>> public class Comparer : IComparer<int[]>
>>> {
>>> public int Compare(int[] x, int[] y)
>>> {
>>> int result = x[0].CompareTo(y[0]);
>>> if (result == 0)
>>> {
>>> result = x[1].CompareTo(y[1]);
>>> }
>>>
>>> return result;
>>> }
>>> }
>>> }
>>> }
>>>
>>> The program comes up with:
>>> 434657, 150568, 420968
>>>
>>> Please reply to ilanlmayer at gmail dot com
>>>
>>> __/\__
>>> \ /
>>> __/\\ //\__ Ilan Mayer
>>> \ /
>>> /__ __\ Toronto, Canada
>>> /__ __\
>>> ||
>> Nice program. The solution found in 1750 was
>>
>> Answered by Mr. C. Bumpkin . By a method of substitution , too tedious to insert , he finds 1873432 , 2399057 , 2288168 , the three numbers , answering the conditions of the question .
>>
>> I did not realize that the name was an alias until prodded by Richard Tobin.
>>
>> L. Flynn
>
> Revised program:
>
> using System;
> using System.Collections.Generic;
>
> namespace SquaresPuzzle
> {
> class SquaresPuzzle
> {
> private const int N = 3000;
>
> static void Main()
> {
> List<int[]> numbers = new List<int[]>();
>
> for (int n1 = 1; n1 <= N; n1++)
> {
> for (int n2 = 1; n2 < n1; n2++)
> {
> if (n1 % 2 == n2 % 2)
> {
> int n1Squared = n1 * n1;
> int n2Squared = n2 * n2;
> int m1 = (n1Squared + n2Squared) / 2;
> int m2 = (n1Squared - n2Squared) / 2;
> numbers.Add(new[] { m1, m2 });
> }
> }
> }
>
> Comparer comparer = new Comparer();
> numbers.Sort(comparer);
> for (int n = 0; n < numbers.Count; n++)
> {
> int m = n;
> while (++m < numbers.Count && numbers[m][0] == numbers[n][0])
> {
> int n1 = numbers[n][1];
> int n2 = numbers[m][1];
> int[] number1 = new[] { n1, n2 };
> int[] number2 = new[] { n2, n1 };
> if (numbers.BinarySearch(number1, comparer) >= 0 || numbers.BinarySearch(number2, comparer) >= 0)
> {
> int n0 = numbers[n][0];
> int divisor1 = FirstDivisor(n0);
> int divisor2 = FirstDivisor(n1);
> int divisor3 = FirstDivisor(n2);
> if (divisor1 != divisor2 || divisor1 != divisor3)
> {
> Console.WriteLine("{0}, {1}, {2}", n0, n1, n2);
> }
> }
> }
> }
> }
>
> private static int FirstDivisor(int number)
> {
> if (number % 2 == 0)
> {
> return 2;
> }
>
> int root = (int) Math.Sqrt(number);
> for (int n = 3; n <= root; n += 2)
> {
> if (number % n == 0)
> {
> return n;
> }
> }
>
> return number;
> }
>
> public class Comparer : IComparer<int[]>
> {
> public int Compare(int[] x, int[] y)
> {
> int result = x[0].CompareTo(y[0]);
> if (result == 0)
> {
> result = x[1].CompareTo(y[1]);
> }
>
> return result;
> }
> }
> }
> }
>
> It produces 5 solutions (excluding solutions where all 3 numbers share a divisor), including the original one:
> 434657, 150568, 420968
> 733025, 418304, 488000
> 2399057, 1873432, 2288168
> 3911913, 1355112, 3788712
> 5320193, 589568, 1782032
>
> Please reply to ilanlmayer at gmail dot com
>

Here is a few more

Solution 88642, 891458, 3713858
Solution 150568, 420968, 434657
Solution 418304, 488000, 733025
Solution 589568, 1782032, 5320193
Solution 602272, 1683872, 1738628
Solution 856350, 949986, 993250
Solution 1355112, 3788712, 3911913
Solution 1673216, 1952000, 2932100
Solution 1761858, 2040642, 2843458
Solution 1873432, 2288168, 2399057
Solution 3425400, 3799944, 3973000

=====
Slightly different program, Square check i+j,j+k are redundant, but I
couldn't be bothered to remove them from the first version that was too
slow.

Java...

package com.company;

public class Main {

public static void main(String[] args) {
doCheck( 10000000);
}

public static void doCheck(long maxtest){
long sqrtMaxtext = (long)Math.floor(Math.sqrt(maxtest)) +1;
for (long i = 1; i < maxtest; i++) {
long sqrt2i = (long)Math.floor(Math.sqrt(2*i))+1;
for (long sj = sqrt2i; sj < sqrtMaxtext; sj++) {
long j = sj*sj-i;
if (isSquare(i + j) && isSquare(j - i)) {
long sqrt2j = (long)Math.floor(Math.sqrt(2*j))+1;
for (long sk = sqrt2j; sk < sqrtMaxtext; sk++) {
long k= sk*sk-j;
if (isSquare(k + i)
&& isSquare(k - i)
&& isSquare(k + j)
&& isSquare(k - j)
) {


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