{2 m n, m^2 - n^2, m^2 + n^2}
is from this formula on wiki and you can easily verify that
$(2mn)^2 + (m^2-n^2)^2 = (m^2 + n^2)^2$. The second part of the code is kind of clever way to generate
- Same pythagorean triples up to common factor
- Eleminate
$m^2 + n^2$'s larger than the
lim
Remove the last part of genPTunder gives:
genPTunder2[lim_Integer?Positive] :=
Module[{prim},
prim = Join @@
Table[If[
CoprimeQ[m, n], {2 m n, m^2 - n^2, m^2 + n^2}, ## &[]], {m, 2,
Floor@Sqrt@lim}, {n, 1 + m~Mod~2, m, 2}]]
Notice that we need to add 2*{3,4,5}, 3*{3,4,5}, 4*{3,4,5} and remove {24,7,25} if we set lim to 20. And we also want to have each tuple sorted numertically:
In[9]:= data = genPTunder2[20]
Out[9]= {{4, 3, 5}, {12, 5, 13}, {8, 15, 17}, {24, 7, 25}}
Quotient function generate the common factors that can be used for multiplication of each triple:
In[11]:= {Range[20~Quotient~Max@#], Sort[#]} & /@ data
Out[11]= {{{1, 2, 3, 4}, {3, 4, 5}}, {{1}, {5, 12, 13}}, {{1}, {8, 15,
17}}, {{}, {7, 24, 25}}}
$20/5 = 4$ therefore {3, 4, 5} can be multiplied up to four times so that the hypothenus is still no larger than 20. Similiarly
$\lfloor 20/13 \rfloor = 1$, {5,12,13} is the only case;
$\lfloor 20/25 \rfloor = 0$ so Range[20~Quotient~25] returns empty list.
KroneckerProduct works like map each multiplier to a given triple and the empty list is eaten by Union in the end:
In[13]:= KroneckerProduct[{1, 2, 3, 4}, {3, 4, 5}]
Out[13]= {{3, 4, 5}, {6, 8, 10}, {9, 12, 15}, {12, 16, 20}}
In[14]:= KroneckerProduct[{}, {7, 24, 25}]
Out[14]= {}
Because most of the funtion used here are either optimized for integer calculation, automatic multithreading or linear algebra operation, this SE implementation is very efficient for large number
