27529 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 148^2 + 75^2.

27529 is a member of OEIS A358744: first of three consecutive primes p, q, r such that p + q - r, p^2 + q^2 - r^2 and p^3 + q^3 - r^3 are all prime.

The sequence can be generated as follows (permalink):

INPUT

L=[]

for p in prime_range(40000):

  q=next_prime(p)

  r=next_prime(q)

  if is_prime(p+q-r):

    if is_prime(p^2+q^2-r^2):

      if is_prime(p^3+q^3-r^3):

        L.append(p)

print(L)

OUTPUT

[13, 29, 137, 521, 577, 691, 823, 1879, 3469, 4799, 8783, 21569, 25453, 26263, 26591, 27529, 27919, 34607, 39509]

27529 cannot be rendered as a digit equation:

*NO_EQUATION*