Finding complete squares summing up to a prime for your own large prime

References for Taster lecture

(University of Exeter, Jan Sieber)

Simon Singh: Fermat's Last Theorem
Wikipedia
The theorem discussed in the lecture is a less famous but still interesting observation also made by Fermat:

Every prime number p that is one above a multiple of 4 (p=4k+1 for an integer k) is the sum of two whole squares.
The Wikipedia page with different historical proofs of this statement is here:
Wikipedia.
How to find out if p is a prime number without trying factors (Miller-Rabin test):
Wikipedia.
Approximately how many prime numbers have at most d digits?
Wikipedia
Approximately how many numbers with at most d digits can be written as the sum of two complete squares?
Wikipedia
Is the splitting of the prime p=4k+1 into a^2+b^2 unique?
math.stackexchange.
See CoCalc Sagemath page for further explanations.

Finding complete squares summing up to a prime for your own large prime

The part below uses the mathematical programming environment sage (www.sagemath.org/). The cells with the numbered lines are programming code, which will be executed on the free cloud service sagecell.sagemath.org/

Type in a large whole number p0 below (replace the 987654321 for example with 2^100) and then click on evaluate. This may take a few seconds as the sagecell server needs to start up a computation.
The following block finds the next prime number p after p0 that has the form p=4k+1. This may take a while for large numbers p0. Click Evaluate to run the code.
The following block finds the whole numbers a and b such that the complete squares a^2 and b^2 add up to p. This should be faster than finding the large prime. Click Evaluate to run the code.