Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss
Groups keyboard shortcuts have been updated
Dismiss
See shortcuts

Primitive Pythagorean triangles of the same area

137 views
Skip to first unread message

zak

unread,
Jan 18, 2005, 4:04:28 PM1/18/05
to
These three primitive Pythagorean triangles
have the same area
{area, lesser leg, larger leg, hypotenuse}
{13123110,1380,19019,19069},
{13123110,3059,8580,9109},
{13123110,4485,5852,7373}.
Are any other such triples, 4,5-tuples?

Thanks,
Zak

Gerry Myerson

unread,
Jan 18, 2005, 4:53:01 PM1/18/05
to
In article <1106082268....@c13g2000cwb.googlegroups.com>,
"zak" <seid...@yahoo.com> wrote:

This is part of Problem D21 in Guy, Unsolved Problems in Number Theory.
The triple above was found by Charles L Shedd in 1945.
In 1986, Rathbun found three more, one of which has generators
(1610, 869), (2002, 1817), (2622, 143) [The triangle with generators
(a, b) is 2ab, a^2 - b^2, a^2 + b^2]. A 5th triple was found
independently by Hoey and Rathbun.

It appears to be unknown whether there is an infinity of such triples,
also whether there are any quadruples.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

zak

unread,
Jan 19, 2005, 6:21:01 AM1/19/05
to
Gerry,
thanks for help!
Can you please provide other known triples
(or links to sites -
I've searched net some but failed).
Also Guy's book is not available to me :=(

Gerry Myerson

unread,
Jan 19, 2005, 5:55:40 PM1/19/05
to
In article <1106133661.1...@z14g2000cwz.googlegroups.com>,
"zak" <seid...@yahoo.com> wrote:

77, 38; 78, 55; 138, 5.
1610, 869; 2002, 1817; 2622, 143.
2035, 266; 3306, 61; 3422, 55.
2201, 1166; 2438, 2035; 3565, 198.
7238, 2465; 9077, 1122; 10434, 731.

(For anyone coming in late, each pair a, b expands to
a primitive Pythagorean triple, 2ab, a^2 - b^2, a^2 + b^2;
the three in each line above have the same value for 2ab(a^2 - b^2)
[hence, the same area when interpreted as right triangles];
these are the five triples of triples listed in D21 of Guy's
Unsolved Problems in Number Theory)

anonymous

unread,
Jan 20, 2005, 3:26:55 AM1/20/05
to

It is difficult to find the 6th one, as the area searched is out to 10^21.
Using Paul Whitlock's parametrization to simplify the search we find:

n,m = (5,6) Found 3 - (138,5) gcd:1 (78,55) gcd:1 (77,38) gcd:1

m,n = (8,3) Found 3 - (88,3) gcd:1 (55,13) gcd:1 (51,40) gcd:1

m,n = (16,7) Found 4 - (368,35) gcd:1 (299,259) gcd:1 (259,144) gcd:1 (259,155)
gcd:1

n,m = (22,31) Found 3 - (3565,198) gcd:1 (2438,2035) gcd:1 (2201,1166) gcd:1

m,n = (32,1) Found 3 - (1056,29) gcd:1 (992,35) gcd:1 (553,377) gcd:1

m,n = (46,11) Found 3 - (2622,143) gcd:1 (2002,1817) gcd:1 (1610,869) gcd:1

m,n = (58,1) Found 3 - (3422,55) gcd:1 (3306,61) gcd:1 (2035,266) gcd:1

m,n = (94,17) Found 3 - (10434,731) gcd:1 (9077,1122) gcd:1 (7238,2465) gcd:1

m,n = (146,97) Found 3 - (45114,41895) gcd:3 (42389,7154) gcd:1 (35478,14065)
gcd:1

n,m = (256,465) Found 3 - (767715,53504) gcd:1 (529635,184576) gcd:1
(506922,236282) gcd:2

n,m = (14,3125) Found 3 - (29340625,43554) gcd:1 (29253125,43946) gcd:1
(19664975,144670) gcd:5

m,n = (3632,3397) Found 4 - (46956731,853520) gcd:1 (25529328,22280923) gcd:1
(22280923,10002493) gcd:1 (22280923,15526835) gcd:1

The m,n search has been carried out to m,n > 20,000,19999 with no triples found

If a quadruple of primitive triangles exist, they must be extremely rare.

Randall


Gerry Myerson

unread,
Jan 20, 2005, 6:52:55 PM1/20/05
to
In article <11062096...@news-1.nethere.net>,
anonymous <somewhere@on_earth.in_space> wrote:

> > In article <1106082268....@c13g2000cwb.googlegroups.com>,
> > "zak" <seid...@yahoo.com> wrote:
> >
> >> These three primitive Pythagorean triangles
> >> have the same area
> >> {area, lesser leg, larger leg, hypotenuse}
> >> {13123110,1380,19019,19069},
> >> {13123110,3059,8580,9109},
> >> {13123110,4485,5852,7373}.
> >> Are any other such triples, 4,5-tuples?
> >
> > This is part of Problem D21 in Guy, Unsolved Problems in Number Theory.
> > The triple above was found by Charles L Shedd in 1945.
> > In 1986, Rathbun found three more, one of which has generators
> > (1610, 869), (2002, 1817), (2622, 143) [The triangle with generators
> > (a, b) is 2ab, a^2 - b^2, a^2 + b^2]. A 5th triple was found
> > independently by Hoey and Rathbun.
> >
> > It appears to be unknown whether there is an infinity of such triples,
> > also whether there are any quadruples.
> >
>
> It is difficult to find the 6th one, as the area searched is out to 10^21.
> Using Paul Whitlock's parametrization to simplify the search we find:

I don't know what Paul Whitlock's parametrization is. I do know
(you do too, of course) that if a and b are both odd then the resulting
triangle isn't primitive, as all three sides have even length. Some
of the entries on your list, namely,

> m,n = (8,3) Found 3 - (88,3) gcd:1 (55,13) gcd:1 (51,40) gcd:1
>
> m,n = (16,7) Found 4 - (368,35) gcd:1 (299,259) gcd:1 (259,144) gcd:1
> (259,155)
> gcd:1
>

> m,n = (32,1) Found 3 - (1056,29) gcd:1 (992,35) gcd:1 (553,377) gcd:1
>

> m,n = (3632,3397) Found 4 - (46956731,853520) gcd:1 (25529328,22280923) gcd:1
> (22280923,10002493) gcd:1 (22280923,15526835) gcd:1

suffer from this defect, so the gcd:1 fooled me for a while.

anonymous

unread,
Jan 29, 2005, 11:31:17 PM1/29/05
to
> I don't know what Paul Whitlock's parametrization is. I do know
> (you do too, of course) that if a and b are both odd then the resulting
> triangle isn't primitive, as all three sides have even length. Some
> of the entries on your list, namely,
>
>> m,n = (8,3) Found 3 - (88,3) gcd:1 (55,13) gcd:1 (51,40) gcd:1
>>
>> m,n = (16,7) Found 4 - (368,35) gcd:1 (299,259) gcd:1 (259,144) gcd:1
>> (259,155)
>> gcd:1
>>
>> m,n = (32,1) Found 3 - (1056,29) gcd:1 (992,35) gcd:1 (553,377) gcd:1
>>
>> m,n = (3632,3397) Found 4 - (46956731,853520) gcd:1 (25529328,22280923) gcd:1
>> (22280923,10002493) gcd:1 (22280923,15526835) gcd:1
>
> suffer from this defect, so the gcd:1 fooled me for a while.

mea culpa. Yes, this I do know well, I should have put in a remark on the
posting, so no one would have been misled.

Thanks for the admonishment.

Randall

duncan...@gmx.com

unread,
Mar 2, 2017, 8:02:09 AM3/2/17
to
On Thursday, 20 January 2005 08:26:55 UTC, anonymous wrote:
> >
> >> These three primitive Pythagorean triangles
> >> have the same area
> >> {area, lesser leg, larger leg, hypotenuse}
> >> {13123110,1380,19019,19069},
> >> {13123110,3059,8580,9109},
> >> {13123110,4485,5852,7373}.
> >> Are any other such triples, 4,5-tuples?
> >
> > This is part of Problem D21 in Guy, Unsolved Problems in Number Theory.
> > The triple above was found by Charles L Shedd in 1945.
> > In 1986, Rathbun found three more, one of which has generators
> > (1610, 869), (2002, 1817), (2622, 143) [The triangle with generators
> > (a, b) is 2ab, a^2 - b^2, a^2 + b^2]. A 5th triple was found
> > independently by Hoey and Rathbun.
> >
> > It appears to be unknown whether there is an infinity of such triples,
> > also whether there are any quadruples.

Using an algorithm which searches exhaustively over areas I've discovered the 6th triple of primitive Pythagorean triangles with equal area. The area is 9381843970167926138271390 and the generators are

(352538,2999447), (1931103,2398838) and (3063347,3215070)

The search took roughly 35GHz-days.

Duncan Moore

bassam king karzeddin

unread,
Mar 2, 2017, 8:38:00 AM3/2/17
to
Congratulation, this is fantastic first post to start with, (DISCOVERY)

My regards
Bassam King Karzeddin
02/03/17

abu.ku...@gmail.com

unread,
Mar 2, 2017, 9:43:51 PM3/2/17
to
I don't really comprehend it, either, but it is c00l

bassam king karzeddin

unread,
Mar 4, 2017, 4:20:20 AM3/4/17
to
This is real mathematics but not cool, because it is INTEGERS ONLY, where one is not allowed to go around the truth (by APPROXIMATION) as MANY other brancheS in mathematics

And if it is that cool, get the next triplet then! Wonder!


BK

abu.ku...@gmail.com

unread,
Mar 4, 2017, 6:00:37 PM3/4/17
to
beat a dead horse, bother the flies
0 new messages