Make {x^2 + xy + y^2, x^2 + xz + z^2, y^2 + yz + z^2} squares

 
 
Can you find 3 distinct rational numbers   x, \; y, \; z   to can make the three expressions squares

x^2 \; + \; x \, y \; + \; y^2
x^2 \; + \; x \, z \; + \; z^2
y^2 \; + \; y \, z \; + \; z^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Primitive Pythagorean triples (a, b=a+1, c)

 
 
Let P be the perimeter

a, a+1, c 1
 
(1)
 
Can you explain the following:

The generators   m   and   n     of the first few triangles are the Pell numbers
2,   5,   12,   29, …,   13860,   and   1,   2,   5, …,   5741,   respectively.

The lengths of their hypotenuses are the Pell numbers,   5,   29,   169,   …
 

A000129
1,   2,   5,   12,   29,   70,   169,   408,   985,   2378,   5741,   13860,   33461,   80782,   195025,   470832,   1136689,   2744210,   6625109,   15994428,   38613965,   93222358,   225058681,   543339720,   1311738121,   3166815962,   7645370045,   18457556052,   44560482149,   107578520350,   259717522849,   …
 
PELL PPT 1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares – Part 2

 
 
Find three positive integers   (x, y, z)   to can make the two expressions squares

    x^2 \; + \; y^2 \; + \; z^2
x^2 \, y^2 \; + \; x^2 \, z^2 \; + \; y^2 \, z^2

 

Here’s a parametric solution established by Euler:

x \; = \; (n^4 - 6 \, n^2 + 1) \,(n^2 + 1)
y \; = \; 4 \, n \, (n^2 - 1)^2
z \; = \; (8 \, n^2) \,(n^2 - 1)

x^2 + y^2 + z^2 = (n^2+1)^6
x^2 \, y^2 + x^2 \, z^2 + y^2 \, z^2 = 16 \, (n-1)^2 \, n^2 \, (n+1)^2 \, (n^8-4 \, n^6+22 \, n^4-4 \, n^2+1)^2

 

xy+xz+yz SQ 2

In two ways:

2881585^2 \; + \; 981552^2 \; + \; 164736^2 \; = \; 145^6
2964815^2 \; + \; 705024^2 \; + \; 83232^2 \; = \; 145^6

 
xy+xz+yz SQ 3

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares — Part 1

 
 
Find three positive integers   (x, y, z)   to can make the two expressions squares

                                              x^2 \; + \; y^2 \; + \; z^2
                                           x^2 \, y^2 \; + \; x^2 \, z^2 \; + \; y^2 \, z^2

 
 
Here are some solutions:

xy+xz+yz SQ 1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {2x^2 + y^2 + z^2, x^2 + 2y^2 + z^2, x^2 + y^2 + 2z^2} squares

 
 
Find three positive integers   (x, y, z)   to can make the three expressions squares

x^2+y^2+z^2 1

 
 

Can you find three distinct positive integers that satisfy the system of equations?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {2x^2 + 2y^2 – z^2, 2x^2 – y^2 + 2z^2, -x^2 + 2*y^2 + 2z^2} squares

 
 

we can make the three expressions squares

(1)    2 \, x^2 \; + \; 2 \, y^2 \; - \; z^2
(2)    2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2
(3)    -x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2

with a simple parameterization:

x \; = \; m \; - \; n
y \; = \; m \; + \; 2 \, n
z \; = \; 2 \, m \; + \; n

(1)
2 \, x^2 \; + 2 \, y^2 \; - \; z^2
= \; 2 \,(m - n)^2 \; + \; 2 \,(m + 2 \, n)^2 \; - \; (2 \, m + n)^2 \; = \; 9 \, n^2

(2)
2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2
= \; 2 \,(m - n)^2 - (m + 2 \, n)^2 + 2 \,(2 \, m + n)^2 \; = \; 9 \, m^2

(3)
-x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2
= \; -(m - n)^2 + 2 \,(m + 2 \, n)^2 \; + \; 2 \,(2 \, m + n)^2 \; = \; 9 \, (m + n)^2

Interestingly,

(2) + (1)
(2 \, x^2 \; - \; y^2 \; + \; 2 \, z^2) \; + \; (2 \, x^2 \; + \; 2 \, y^2 \; - \; z^2) \; = \; 9 \, (m^2 + n^2)

(3)
-x^2 \; + \; 2 \, y^2 \; + \; 2 \, z^2 \; = \; 9 \, (m + n)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {x^2 + y*z, y^2 + x*z, z^2 + x*y} squares

 
 
Find three distinct positive integers   (x, y, z)   to make the three expressions squares
 

Here are some solutions

x^2+yz 1

 
 
Find other solutions
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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When the product of any two of (a,b,c,d) is one less an integer squared

 
 
a, \; b, \; c, \; d   are positive integers
 

a simple parameterization

a \; = \; n \; - \; 1
b \; = \; n \; + \; 1
c \; = \; 4 \, n
d \; = \; 4 \, n \, (4 \,n^2 - 1)

 

1 \; + \; a \, b \; = \; 1 \; + \; (n - 1) \,(n + 1) \; = \; n^2

1 \; + \; a \, c \; = \; 1 \; + \; 4 \, n \,(n - 1) \; = \; (2 n - 1)^2

1 \; + \; b \, c \; = \; 1 \; + \; 4 \, n \,(n + 1) \; = \; (2 n + 1)^2

1 \; + \; a \, d \; = \; 1 \; + \; (n - 1) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (4 \, n^2 - 2 \, n - 1)^2

1 \; + \; b \, d \; = \; 1 \; + \; (n + 1) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (4 \, n^2 + 2 \, n - 1)^2

1 \; + \; c \, d \; = \; 1 \; + \; (4 \, n) \,(4 \, n) \,(4 \, n^2 - 1) \; = \; (8 \, n^2 - 1)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {a^2 + b, a + b^2} squares

 
 
Find two distinct rational numbers   (a, \; b)   to make the two of the expressions squares

                                                 a^2 \; + \; b
                                                 a \; + \; b^2

 

Here are some solutions

x^2+y 1
x^2+y 2

 
 

Steve Kass found a pair   (144/299,  155/299)   where   M = N

(144/299)^2 \; + \; (155/299) \; = \; (144/299) \; + \; (155/299)^2 \; = \; (259/299)^2

 

I found another pair :   (3/8,   5/8)

(3/8)^2 \; + \; (5/8) \; = \; (3/8) \; + \; (5/8)^2 \; = \; (7/8)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Make {a^2 b^2 + c^2 d^2, a^2 c^2 + b^2 d^2, a^2 d^2 + b^2 c^2} squares

 
 
Find positive integers   a, \; b, \; c, \; d   such that

a^2 \, b^2 \; + \; c^2 \, d^2
a^2 \, c^2 \; + \; b^2 \, d^2
a^2 \, d^2 \; + \; b^2 \, c^2

are made to be squares

 

(a, b, c, d) = 1

for   a \; \leq \; 100
 

(a,   b,   c,   d)   =   (3,   14,   8,   5)

(3^2) \,(14^2) \; + \; (8^2) \,(5^2) \; = \; 58^2
(3^2) \,(8^2) \; + \; (14^2) \,(5^2) \; = \; 74^2
(3^2) \,(5^2) \; + \; (14^2) \,(8^2) \; = \; 113^2

(a,   b,   c,   d)   =   (5,   38,   21,   16)

(5^2) \,(38^2) \; + \; (21^2) \,(16^2) \; = \; 386^2
(5^2) \,(21^2) \; + \; (38^2) \,(16^2) \; = \; 617^2
(5^2) \,(16^2) \; + \; (38^2) \,(21^2) \; = \; 802^2

(a,   b,   c,   d)   =   (8,   14,   5,   3)

(8^2) \,(14^2) \; + \; (5^2) \,(3^2) \; = \; 113^2
(8^2) \,(5^2) \; + \; (14^2) \,(3^2) \; = \; 58^2
(8^2) \,(3^2) \; + \; (14^2) \,(5^2) \; = \; 74^2

(a,   b,   c,   d)   =   (8,   26,   15,   7)

(8^2) \,(26^2) \; + \; (15^2) \,(7^2) \; = \; 233^2
(8^2) \,(15^2) \; + \; (26^2) \,(7^2) \; = \; 218^2
(8^2) \,(7^2) \; + \; (26^2) \,(15^2) \; = \; 394^2

(a,   b,   c,   d)   =   (11,   62,   35,   24)

(11^2) \,(62^2) \; + \; (35^2) \,(24^2) \; = \; 1082^2
(11^2) \,(35^2) \; + \; (62^2) \,(24^2) \; = \; 1537^2
(11^2) \,(24^2) \; + \; (62^2) \,(35^2) \; = \; 2186^2

(a,   b,   c,   d)   =   (13,   86,   48,   35)

(13^2) \,(86^2) \; + \; (48^2) \,(35^2) \; = \; 2018^2
(13^2) \,(48^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2
(13^2) \,(35^2) \; + \; (86^2) \,(48^2) \; = \; 4153^2

(a,   b,   c,   d)   =   (15,   26,   7,   8)

(15^2) \,(26^2) \; + \; (7^2) \,(8^2) \; = \; 394^2
(15^2) \,(7^2) \; + \; (26^2) \,(8^2) \; = \; 233^2
(15^2) \,(8^2) \; + \; (26^2) \,(7^2) \; = \; 218^2

(a,   b,   c,   d)   =   (15,   26,   8,   7)

(15^2) \,(26^2) \; + \; (8^2) \,(7^2) \; = \; 394^2
(15^2) \,(8^2) \; + \; (26^2) \,(7^2) \; = \; 218^2
(15^2) \,(7^2) \; + \; (26^2) \,(8^2) \; = \; 233^2

(a,   b,   c,   d)   =   (16,   38,   21,   5)

(16^2) \,(38^2) \; + \; (21^2) \,(5^2) \; = \; 617^2
(16^2) \,(21^2) \; + \; (38^2) \,(5^2) \; = \; 386^2
(16^2) \,(5^2) \; + \; (38^2) \,(21^2) \; = \; 802^2

(a,   b,   c,   d)   =   (21,   38,   5,   16)

(21^2) \,(38^2) \; + \; (5^2)(16^2) \; = \; 802^2
(21^2) \,(5^2) \; + \; (38^2)(16^2) \; = \; 617^2
(21^2) \,(16^2) \; + \; (38^2)(5^2) \; = \; 386^2

(a,   b,   c,   d)   =   (24,   62,   35,   11)

(24^2) \,(62^2) \; + \; (35^2) \,(11^2) \; = \; 1537^2
(24^2) \,(35^2) \; + \; (62^2) \,(11^2) \; = \; 1082^2
(24^2) \,(11^2) \; + \; (62^2) \,(35^2) \; = \; 2186^2

(a,   b,   c,   d)   =   (32,   134,   77,   45)

(32^2) \,(134^2) \; + \; (77^2) \,(45^2) \; = \; 5513^2
(32^2) \,(77^2) \; + \; (134^2) \,(45^2) \; = \; 6514^2
(32^2) \,(45^2) \; + \; (134^2) \,(77^2) \; = \; 10418^2

(a,   b,   c,   d)   =   (33,   74,   40,   7)

(33^2) \,(74^2) \; + \; (40^2) \,(7^2) \; = \; 2458^2
(33^2) \,(40^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2
(33^2) \,(7^2) \; + \; (74^2) \,(40^2) \; = \; 2969^2

(a,   b,   c,   d)   =   (35,   62,   11,   24)

(35^2) \,(62^2) \; + \; (11^2) \,(24^2) \; = \; 2186^2
(35^2) \,(11^2) \; + \; (62^2) \,(24^2) \; = \; 1537^2
(35^2) \,(24^2) \; + \; (62^2) \,(11^2) \; = \; 1082^2

(a,   b,   c,   d)   =   (39,   98,   55,   16)

(39^2) \,(98^2) \; + \; (55^2) \,(16^2) \; = \; 3922^2
(39^2) \,(55^2) \; + \; (98^2) \,(16^2) \; = \; 2657^2
(39^2) \,(16^2) \; + \; (98^2) \,(55^2) \; = \; 5426^2

(a,   b,   c,   d)   =   (40,   74,   7,   33)

(40^2) \,(74^2) \; + \; (7^2) \,(33^2) \; = \; 2969^2
(40^2) \,(7^2) \; + \; (74^2) \,(33^2) \; = \; 2458^2
(40^2) \,(33^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2

(a,   b,   c,   d)   =   (40,   74,   33,   7)

(40^2) \,(74^2) \; + \; (33^2) \,(7^2) \; = \; 2969^2
(40^2) \,(33^2) \; + \; (74^2) \,(7^2) \; = \; 1418^2
(40^2) \,(7^2) \; + \; (74^2) \,(33^2) \; = \; 2458^2

(a,   b,   c,   d)   =   (48,   86,   13,   35)

(48^2) \,(86^2) \; + \; (13^2) \,(35^2) \; = \; 4153^2
(48^2) \,(13^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2
(48^2) \,(35^2) \; + \; (86^2) \,(13^2) \; = \; 2018^2

(a,   b,   c,   d)   =   (48,   86,   35,   13)

(48^2) \,(86^2) \; + \; (35^2) \,(13^2) \; = \; 4153^2
(48^2) \,(35^2) \; + \; (86^2) \,(13^2) \; = \; 2018^2
(48^2) \,(13^2) \; + \; (86^2) \,(35^2) \; = \; 3074^2

(a,   b,   c,   d)   =   (55,   98,   39,   16)

(55^2) \,(98^2) \; + \; (39^2) \,(16^2) \; = \; 5426^2
(55^2) \,(39^2) \; + \; (98^2) \,(16^2) \; = \; 2657^2
(55^2) \,(16^2) \; + \; (98^2) \,(39^2) \; = \; 3922^2

(a,   b,   c,   d)   =   (56,   122,   65,   9)

(56^2) \,(122^2) \; + \; (65^2) \,(9^2) \; = \; 6857^2
(56^2) \,(65^2) \; + \; (122^2) \,(9^2) \; = \; 3802^2
(56^2) \,(9^2) \; + \; (122^2) \,(65^2) \; = \; 7946^2

(a,   b,   c,   d)   =   (65,   122,   56,   9)

(65^2) \,(122^2) \; + \; (56^2) \,(9^2) \; = \; 7946^2
(65^2) \,(56^2) \; + \; (122^2) \,(9^2) \; = \; 3802^2
(65^2) \,(9^2) \; + \; (122^2) \,(56^2) \; = \; 6857^2

(a,   b,   c,   d)   =   (77,   134,   32,   45)

(77^2) \,(134^2) \; + \; (32^2) \,(45^2) \; = \; 10418^2
(77^2) \,(32^2) \; + \; (134^2) \,(45^2) \; = \; 6514^2
(77^2) \,(45^2) \; + \; (134^2) \,(32^2) \; = \; 5513^2

(a,   b,   c,   d)   =   (80,   146,   17,   63)

(80^2) \,(146^2) \; + \; (17^2) \,(63^2) \; = \; 11729^2
(80^2) \,(17^2) \; + \; (146^2) \,(63^2) \; = \; 9298^2
(80^2) \,(63^2) \; + \; (146^2) \,(17^2) \; = \; 5618^2

(a,   b,   c,   d)   =   (80,   146,   63,   17)

(80^2) \,(146^2) \; + \; (63^2) \,(17^2) \; = \; 11729^2
(80^2) \,(63^2) \; + \; (146^2) \,(17^2) \; = \; 5618^2
(80^2) \,(17^2) \; + \; (146^2) \,(63^2) \; = \; 9298^2

(a,   b,   c,   d)   =   (80,   182,   99,   19)

(80^2) \,(182^2) \; + \; (99^2) \,(19^2) \; = \; 14681^2
(80^2) \,(99^2) \; + \; (182^2) \,(19^2) \; = \; 8642^2
(80^2) \,(19^2) \; + \; (182^2) \,(99^2) \; = \; 18082^2

(a,   b,   c,   d)   =   (88,   266,   153,   65)

(88^2) \,(266^2) \; + \; (153^2) \,(65^2) \; = \; 25433^2
(88^2) \,(153^2) \; + \; (266^2) \,(65^2) \; = \; 21914^2
(88^2) \,(65^2) \; + \; (266^2) \,(153^2) \; = \; 41098^2

(a,   b,   c,   d)   =   (91,   158,   51,   40)

(91^2) \,(158^2) \; + \; (51^2) \,(40^2) \; = \; 14522^2
(91^2) \,(51^2) \; + \; (158^2) \,(40^2) \; = \; 7841^2
(91^2) \,(40^2) \; + \; (158^2) \,(51^2) \; = \; 8842^2

(a,   b,   c,   d)   =   (96,   182,   11,   85)

(96^2) \,(182^2) \; + \; (11^2) \,(85^2) \; = \; 17497^2
(96^2) \,(11^2) \; + \; (182^2) \,(85^2) \; = \; 15506^2
(96^2) \,(85^2) \; + \; (182^2) \,(11^2) \; = \; 8402^2

(a,   b,   c,   d)   =   (96,   182,   85,   11)

(96^2) \,(182^2) \; + \; (85^2) \,(11^2) \; = \; 17497^2
(96^2) \,(85^2) \; + \; (182^2) \,(11^2) \; = \; 8402^2
(96^2) \,(11^2) \; + \; (182^2) \,(85^2) \; = \; 15506^2

(a,   b,   c,   d)   =   (99,   182,   19,   80)

(99^2) \,(182^2) \; + \; (19^2) \,(80^2) \; = \; 18082^2
(99^2) \,(19^2) \; + \; (182^2) \,(80^2) \; = \; 14681^2
(99^2) \,(80^2) \; + \; (182^2) \,(19^2) \; = \; 8642^2

 
 
 

Find other solutions.
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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