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edited Jul 25, 2016 at 19:27

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Proof of "if$ a^2"if $a^2 + b^2 = c^2$ then $abc$ is divisible by 60"

Sorry if this question is too simple.

I once read, on a number theory textbook --- forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

a = p^2 - q^2 b = 2pq c = p^2 + q^2\begin{align} a &= p^2 - q^2 \\\ b &= 2pq \\\ c &= p^2 + q^2. \end{align}

I proved it by proving all possible parities of p$p$ and q$q$. It's tedious because I have to prove some cases are not possible (like a$a$, b$b$, and c$c$ can't be all even or odd).

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? usingUsing Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.

Proof of "if$ a^2 + b^2 = c^2$ then $abc$ is divisible by 60"

Sorry if this question is too simple.

I once read, on a number theory textbook --- forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

a = p^2 - q^2 b = 2pq c = p^2 + q^2

I proved it by proving all possible parities of p and q. It's tedious because I have to prove some cases are not possible (like a, b, and c can't be all even or odd)

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? using Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.

Proof of "if $a^2 + b^2 = c^2$ then $abc$ is divisible by 60"

Sorry if this question is too simple.

I once read, on a number theory textbook - forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

\begin{align} a &= p^2 - q^2 \\\ b &= 2pq \\\ c &= p^2 + q^2. \end{align}

I proved it by proving all possible parities of $p$ and $q$. It's tedious because I have to prove some cases are not possible (like $a$, $b$, and $c$ can't be all even or odd).

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? Using Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.

added the dollar signs

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edit approved Jul 25, 2016 at 19:27

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Proof of "if"if$ a^2 + b^2 = c^2c^2$ then abc$abc$ is divisible by 60"

Hi,

Sorry if this question is too simple.

I once read, on a number theory textbook --- forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

a = p^2 - q^2 b = 2pq c = p^2 + q^2

I proved it by proving all possible parities of p and q. It's tedious because I have to prove some cases are not possible (like a, b, and c can't be all even or odd)

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? using Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.

Proof of "if a^2 + b^2 = c^2 then abc is divisible by 60"

Hi,

Sorry if this question is too simple.

I once read, on a number theory textbook --- forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

a = p^2 - q^2 b = 2pq c = p^2 + q^2

I proved it by proving all possible parities of p and q. It's tedious because I have to prove some cases are not possible (like a, b, and c can't be all even or odd)

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? using Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.

Proof of "if$ a^2 + b^2 = c^2$ then $abc$ is divisible by 60"

Sorry if this question is too simple.

I once read, on a number theory textbook --- forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60.

I proved that using the generating functions (is this the correct name? I got the name from my Discrete Mathematics textbook):

a = p^2 - q^2 b = 2pq c = p^2 + q^2

I proved it by proving all possible parities of p and q. It's tedious because I have to prove some cases are not possible (like a, b, and c can't be all even or odd)

My questions are:

Who and how someone came up with the generating functions?
If you don't know the generating functions or don't want to prove it like I did, is there any other way to prove it? Geometrically? using Calculus? I mean there're many ways to prove Pythagorean theorem using Geometry, Number Theory, etc.