27380 is a product of a power of 2, a 4k+1 prime (5) and a 4k+1 prime (37) raised to the power of 2. Thus it can be expressed as a sum of two squares in three different ways viz. 22^2+164^2, 74^2+148^2 and 116^2+118^2.

27380 is a member of OEIS A257547: even side of integer-sided triangle such that the area is integer (here 3203694) & two sides are twin primes (here 13691 & 13693). The sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2)=(p, p+2,q) where p and p+2 primes. Note that q = 2*p - 2. Note that the angle opposite the even side is very close to 90 degrees as the perpendicular height of this triangle is only about 234 units (permalink).

INPUT

q=27380

p=(q+2)/2

s=(q+p+p+2)/2

A=sqrt(s*(s-q)*(s-p)*(s-(p+2)))

print("The even base of the triangle is",q)

print("The area of the triangle is",A)

print("The twin prime sides are",p,"and",p+2)

print("The perpendicular height of the triangle is about",n(2*A/q,digits=5))

OUTPUT

Even side of triangle is 27380

The area of the triangle is 3203694

The twin prime sides are 13691 and 13693

The perpendicular height of the triangle is about 234.02

27380 is a Harshad, untouchable, practical, abundant, pseudoperfect and Zumkeller number.

27380 can be expressed as a digit equation via 2 = 7 + 3 - 8 + 0.

27380 stabilises after about 380 generations into a single glider plus an assortment of still lifes and oscillators.

There is a hexomino oscillator called "the toad" that I hadn't seen before in this context:

X X X

X X X

There is a still life "honey farm" consisting of four "beehives". There are two still life "blocks", two oscillators called "traffic lights" and a still life called "a loaf".