Solving Fractional Diophantine EquationsQuadratic fields and solving Diophantine equationsSolving a system of Diophantine equationsSolving linear Diophantine equations in 3 variablesSolving quadratic diophantine equations in two variablesSolving two diophantine equations.Solving diophantine equationssolving a Diophantine system of two equationsTechniques for solving Diophantine equations.Approximating a decimal with a fraction (32-bit fixed point to two 23-bit numbers). Think binary, ease of computation.Solving diophantine equations: Finding integer solutions
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Solving Fractional Diophantine Equations
Quadratic fields and solving Diophantine equationsSolving a system of Diophantine equationsSolving linear Diophantine equations in 3 variablesSolving quadratic diophantine equations in two variablesSolving two diophantine equations.Solving diophantine equationssolving a Diophantine system of two equationsTechniques for solving Diophantine equations.Approximating a decimal with a fraction (32-bit fixed point to two 23-bit numbers). Think binary, ease of computation.Solving diophantine equations: Finding integer solutions
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As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$dfrac3-n^22n-12=k$$ To continue, I need to know what integer values of $n$ make $k$ an integer. Since I want this to be a factorization algorithm, I would like methods that don't include factoring; and since I want this to be efficient, I would like methods that doesn't include checking between a range. Other than those two exceptions, any method is accepted, no matter how complicated (I can always optimize it later).
calculus algebra-precalculus diophantine-equations fractions
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
$endgroup$
add a comment |
$begingroup$
As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$dfrac3-n^22n-12=k$$ To continue, I need to know what integer values of $n$ make $k$ an integer. Since I want this to be a factorization algorithm, I would like methods that don't include factoring; and since I want this to be efficient, I would like methods that doesn't include checking between a range. Other than those two exceptions, any method is accepted, no matter how complicated (I can always optimize it later).
calculus algebra-precalculus diophantine-equations fractions
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
$endgroup$
add a comment |
$begingroup$
As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$dfrac3-n^22n-12=k$$ To continue, I need to know what integer values of $n$ make $k$ an integer. Since I want this to be a factorization algorithm, I would like methods that don't include factoring; and since I want this to be efficient, I would like methods that doesn't include checking between a range. Other than those two exceptions, any method is accepted, no matter how complicated (I can always optimize it later).
calculus algebra-precalculus diophantine-equations fractions
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
$endgroup$
As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$dfrac3-n^22n-12=k$$ To continue, I need to know what integer values of $n$ make $k$ an integer. Since I want this to be a factorization algorithm, I would like methods that don't include factoring; and since I want this to be efficient, I would like methods that doesn't include checking between a range. Other than those two exceptions, any method is accepted, no matter how complicated (I can always optimize it later).
calculus algebra-precalculus diophantine-equations fractions
calculus algebra-precalculus diophantine-equations fractions
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
asked Mar 29 at 16:44
Quote DaveQuote Dave
194
194
add a comment |
add a comment |
1 Answer
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Hint: If $A=frac3-n^22n-12$ has to be an integer, so does $2cdot A$
$$=frac-2n^2+62n-12=-n-frac3-6nn-6=-n-frac-6(n-6)-33n-6=-(n+6)-frac33n-6$$ and hence, you're looking for values of $n$, for which
$$frac33n-6inmathbb Z$$
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
$endgroup$
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Hint: If $A=frac3-n^22n-12$ has to be an integer, so does $2cdot A$
$$=frac-2n^2+62n-12=-n-frac3-6nn-6=-n-frac-6(n-6)-33n-6=-(n+6)-frac33n-6$$ and hence, you're looking for values of $n$, for which
$$frac33n-6inmathbb Z$$
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
$endgroup$
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
add a comment |
$begingroup$
Hint: If $A=frac3-n^22n-12$ has to be an integer, so does $2cdot A$
$$=frac-2n^2+62n-12=-n-frac3-6nn-6=-n-frac-6(n-6)-33n-6=-(n+6)-frac33n-6$$ and hence, you're looking for values of $n$, for which
$$frac33n-6inmathbb Z$$
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
$endgroup$
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
add a comment |
$begingroup$
Hint: If $A=frac3-n^22n-12$ has to be an integer, so does $2cdot A$
$$=frac-2n^2+62n-12=-n-frac3-6nn-6=-n-frac-6(n-6)-33n-6=-(n+6)-frac33n-6$$ and hence, you're looking for values of $n$, for which
$$frac33n-6inmathbb Z$$
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
$endgroup$
Hint: If $A=frac3-n^22n-12$ has to be an integer, so does $2cdot A$
$$=frac-2n^2+62n-12=-n-frac3-6nn-6=-n-frac-6(n-6)-33n-6=-(n+6)-frac33n-6$$ and hence, you're looking for values of $n$, for which
$$frac33n-6inmathbb Z$$
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
edited Mar 29 at 17:54
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
answered Mar 29 at 17:39
Dr. MathvaDr. Mathva
3,313630
3,313630
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
add a comment |
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
$begingroup$
Yes but that would require a. Factorization, or b. Checking over a range, both of which I would not like. Are there any other methods?
$endgroup$
– Quote Dave
Mar 29 at 18:10
add a comment |
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