Final answer:
By exploring the possible values of a² and 5b² modulo 20 and their combinations, we can conclude that for a prime number p expressed as a² + 5b², p must be congruent to 1 or 9 modulo 20.
Step-by-step explanation:
To show that a prime p can be written in the form p=a²+5b² implies p≡1 or 9 (mod 20), first assume that p can indeed be expressed as a² + 5b². The values of a² will be congruent to 0, 1, 4, 9, 16 or potentially higher squares when modulo 20, but since we are dealing with a prime, a must not result in 0 when squared modulo 20, so the relevant values for a² are 1, 4, 9, or 16. Similarly, the possible values of 5b² will include the square terms of b, which also must be non-zero, times 5, which will yield congruent results of 0, 5, 20, and so on. When combining these two terms, a² and 5b², the only combinations that result in a prime number congruent to 1 or 9 modulo 20 are where a² is congruent to 1 or 9, and 5b² is congruent to 0 or 20 modulo 20. Therefore, through the possible combinations and the nature of modulo arithmetic, we're left to conclude that the prime number p must indeed be congruent to 1 or 9 modulo 20.