Let L be the total (integer) length of an integer sided right angle triangle. For a given value of L, we can either make no integer sided right triangles. More than one such triangle. Or only one.
For how many values of L <= 1500000 does only one triangle as described exist?
The requre us to to construct all primitive pythagorean triples. That can be done by:
\[a = m^2 - n^2\]
\[b = 2mn\]
\[c = m^2 + n^2\]
Where \(m>n>0\)
We get some limitations:
\[a+b+c < 1500000\]
\[m^2 + 2mn + m^2 < 1500000\]
\[2m^2 + 2mn < 1500000\] \[2(m^2 + mn) < 1500000\] \[(m^2 + mn) < 1500000/2\]
Since \(m>n\), and \(n>0\) we get that \(m^2 + m < 1500000/2\)
And therefore the upper limit of m is 865
Other limits exist. At least one value of m and n have to be even. And they have to be coprime.
Lets identify all m and ns neede for calculating the relevant primitive pythagorean triples:
library(tidyr)
library(numbers)
library(dplyr)
library(purrr)
candidate_mn <- expand.grid(m= 1:866, n = 1:866) %>%
filter(m>n) %>%
mutate(at_least_one_even = 1-m%%2 + 1- n%%2) %>%
filter(at_least_one_even != 0) %>%
select(-at_least_one_even)Filtering for coprime:
candidate_mn <- candidate_mn |>
mutate(coprim = (map2(m,n, coprime))) |>
unnest(coprim) |>
filter(coprim)Generating a,b and c - and filtering for too large values:
candidate_abc <- candidate_mn |>
mutate(a= m^2 - n^2,
b = 2*m*n,
c = m^2 + n^2,
total = a+b+c) %>%
filter(total <= 1500000) A primitive triangle with length 12, can construct triangels with length 12, 24, 36 etc. Generating all multiple of them:
more_candidates <- candidate_abc %>%
mutate(length = map(total, function(x) seq(x, 1500010,x))) |>
unnest(length)Filtering, grouping by length, counting number of times a given length occurs, and only counting the unique:
answer <- more_candidates %>%
select(-c(m,n)) |>
filter(length <= 1500000) |>
group_by(length) %>%
add_tally() %>%
ungroup() %>%
filter(n==1) |>
nrow()