Mixing
What are the positive integral solution sets of $a^2+ab+b^2=343k$?
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I had encountered a problem in which we had to find all positive integers $(a,b,k)$ such that $a^2+ab+b^2=343k$. I had never had a general approach to solve such an equation having three variables. Would someone please help me to solve this equation?
number-theory
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
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add a comment |
$begingroup$
I had encountered a problem in which we had to find all positive integers $(a,b,k)$ such that $a^2+ab+b^2=343k$. I had never had a general approach to solve such an equation having three variables. Would someone please help me to solve this equation?
number-theory
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
$endgroup$
5
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42
add a comment |
$begingroup$
I had encountered a problem in which we had to find all positive integers $(a,b,k)$ such that $a^2+ab+b^2=343k$. I had never had a general approach to solve such an equation having three variables. Would someone please help me to solve this equation?
number-theory
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
$endgroup$
I had encountered a problem in which we had to find all positive integers $(a,b,k)$ such that $a^2+ab+b^2=343k$. I had never had a general approach to solve such an equation having three variables. Would someone please help me to solve this equation?
number-theory
number-theory
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
asked Jan 26 at 17:34
Shashwat1337Shashwat1337
959
959
5
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42
add a comment |
5
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42
5
5
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42
add a comment |
0
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5
$begingroup$
Hint: Write this as $a^3 equiv b^3$ mod 343 and $a neq b$ mod 343. Also $343=7^3.$
$endgroup$
– Aravind
Jan 26 at 17:42