Find all integer solutions of $3(m^2 + n^2) - 7(m+n) = -4$.
The solution and further information about how to solve this type of equation about how to solve this type of lattice point and circles will be much appreciated. Thanks in advance.
circle diophantine-equations
add a comment |
The solution and further information about how to solve this type of equation about how to solve this type of lattice point and circles will be much appreciated. Thanks in advance.
circle diophantine-equations
2
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30
add a comment |
The solution and further information about how to solve this type of equation about how to solve this type of lattice point and circles will be much appreciated. Thanks in advance.
circle diophantine-equations
The solution and further information about how to solve this type of equation about how to solve this type of lattice point and circles will be much appreciated. Thanks in advance.
circle diophantine-equations
circle diophantine-equations
asked Jan 4 at 17:22
Shafin AhmedShafin Ahmed
345
345
2
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30
add a comment |
2
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30
2
2
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30
add a comment |
3 Answers
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First, expand the equation and bring the $-4$ to the other side.
$$3m^2+3n^2-7m-7n+4 = 0$$
To solve for $m$, you can rearrange the LHS:
$$3m^2-7m+left(3n^2-7n+4right) = 0$$
By the Quadratic Formula,
$$m = frac{-(-7)pmsqrt{(-7)^2-4(3)left(3n^2-7n+4right)}}{2(3)}$$
$$m = frac{7pmsqrt{1-36n^2+84n}}{6}$$
You want the discriminant to be a perfect square:
$$1-36n^2+84n = t^2$$
and
$$1-36n^2+84n geq 0$$
add a comment |
Hint:
$$3m^2-7m+3n^2-7n+4=0$$
What is the discriminant of the quadratic equation in $m$
add a comment |
Following the comment by @saulspatz, we get
$$
(6m-7)^2+(6n-7)^2=50
$$
So, we have to solve $a^2+b^2=50$ with $a equiv b equiv -7 bmod 6$.
The solutions are
$$
begin{array}{cccc}
a & b & m & n \
-7 & -1 & 0 & 1 \
-1 & -7 & 1 & 0 \
5 & 5 & 2 & 2 \
end{array}
$$
These were found by brute force. For numbers larger than $50$, consider its prime factorization and use the Brahmagupta–Fibonacci identity.
add a comment |
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3 Answers
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active
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3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
First, expand the equation and bring the $-4$ to the other side.
$$3m^2+3n^2-7m-7n+4 = 0$$
To solve for $m$, you can rearrange the LHS:
$$3m^2-7m+left(3n^2-7n+4right) = 0$$
By the Quadratic Formula,
$$m = frac{-(-7)pmsqrt{(-7)^2-4(3)left(3n^2-7n+4right)}}{2(3)}$$
$$m = frac{7pmsqrt{1-36n^2+84n}}{6}$$
You want the discriminant to be a perfect square:
$$1-36n^2+84n = t^2$$
and
$$1-36n^2+84n geq 0$$
add a comment |
First, expand the equation and bring the $-4$ to the other side.
$$3m^2+3n^2-7m-7n+4 = 0$$
To solve for $m$, you can rearrange the LHS:
$$3m^2-7m+left(3n^2-7n+4right) = 0$$
By the Quadratic Formula,
$$m = frac{-(-7)pmsqrt{(-7)^2-4(3)left(3n^2-7n+4right)}}{2(3)}$$
$$m = frac{7pmsqrt{1-36n^2+84n}}{6}$$
You want the discriminant to be a perfect square:
$$1-36n^2+84n = t^2$$
and
$$1-36n^2+84n geq 0$$
add a comment |
First, expand the equation and bring the $-4$ to the other side.
$$3m^2+3n^2-7m-7n+4 = 0$$
To solve for $m$, you can rearrange the LHS:
$$3m^2-7m+left(3n^2-7n+4right) = 0$$
By the Quadratic Formula,
$$m = frac{-(-7)pmsqrt{(-7)^2-4(3)left(3n^2-7n+4right)}}{2(3)}$$
$$m = frac{7pmsqrt{1-36n^2+84n}}{6}$$
You want the discriminant to be a perfect square:
$$1-36n^2+84n = t^2$$
and
$$1-36n^2+84n geq 0$$
First, expand the equation and bring the $-4$ to the other side.
$$3m^2+3n^2-7m-7n+4 = 0$$
To solve for $m$, you can rearrange the LHS:
$$3m^2-7m+left(3n^2-7n+4right) = 0$$
By the Quadratic Formula,
$$m = frac{-(-7)pmsqrt{(-7)^2-4(3)left(3n^2-7n+4right)}}{2(3)}$$
$$m = frac{7pmsqrt{1-36n^2+84n}}{6}$$
You want the discriminant to be a perfect square:
$$1-36n^2+84n = t^2$$
and
$$1-36n^2+84n geq 0$$
answered Jan 4 at 17:42
KM101KM101
5,7411423
5,7411423
add a comment |
add a comment |
Hint:
$$3m^2-7m+3n^2-7n+4=0$$
What is the discriminant of the quadratic equation in $m$
add a comment |
Hint:
$$3m^2-7m+3n^2-7n+4=0$$
What is the discriminant of the quadratic equation in $m$
add a comment |
Hint:
$$3m^2-7m+3n^2-7n+4=0$$
What is the discriminant of the quadratic equation in $m$
Hint:
$$3m^2-7m+3n^2-7n+4=0$$
What is the discriminant of the quadratic equation in $m$
answered Jan 4 at 17:30
lab bhattacharjeelab bhattacharjee
224k15156274
224k15156274
add a comment |
add a comment |
Following the comment by @saulspatz, we get
$$
(6m-7)^2+(6n-7)^2=50
$$
So, we have to solve $a^2+b^2=50$ with $a equiv b equiv -7 bmod 6$.
The solutions are
$$
begin{array}{cccc}
a & b & m & n \
-7 & -1 & 0 & 1 \
-1 & -7 & 1 & 0 \
5 & 5 & 2 & 2 \
end{array}
$$
These were found by brute force. For numbers larger than $50$, consider its prime factorization and use the Brahmagupta–Fibonacci identity.
add a comment |
Following the comment by @saulspatz, we get
$$
(6m-7)^2+(6n-7)^2=50
$$
So, we have to solve $a^2+b^2=50$ with $a equiv b equiv -7 bmod 6$.
The solutions are
$$
begin{array}{cccc}
a & b & m & n \
-7 & -1 & 0 & 1 \
-1 & -7 & 1 & 0 \
5 & 5 & 2 & 2 \
end{array}
$$
These were found by brute force. For numbers larger than $50$, consider its prime factorization and use the Brahmagupta–Fibonacci identity.
add a comment |
Following the comment by @saulspatz, we get
$$
(6m-7)^2+(6n-7)^2=50
$$
So, we have to solve $a^2+b^2=50$ with $a equiv b equiv -7 bmod 6$.
The solutions are
$$
begin{array}{cccc}
a & b & m & n \
-7 & -1 & 0 & 1 \
-1 & -7 & 1 & 0 \
5 & 5 & 2 & 2 \
end{array}
$$
These were found by brute force. For numbers larger than $50$, consider its prime factorization and use the Brahmagupta–Fibonacci identity.
Following the comment by @saulspatz, we get
$$
(6m-7)^2+(6n-7)^2=50
$$
So, we have to solve $a^2+b^2=50$ with $a equiv b equiv -7 bmod 6$.
The solutions are
$$
begin{array}{cccc}
a & b & m & n \
-7 & -1 & 0 & 1 \
-1 & -7 & 1 & 0 \
5 & 5 & 2 & 2 \
end{array}
$$
These were found by brute force. For numbers larger than $50$, consider its prime factorization and use the Brahmagupta–Fibonacci identity.
answered Jan 4 at 18:02
lhflhf
163k10167388
163k10167388
add a comment |
add a comment |
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2
Multiply both sides by $12$ and then complete squares.
– saulspatz
Jan 4 at 17:30