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Prove that a^t = -1 (mod p^k) for...
p<>2, prime and ord p^k (a) = 2t.
p<>2, prime and ord p^k (a) = 2t.
The equation "a^t = -1 (mod p^k)" means that when the number "a" is raised to the power of "t", and then divided by the number "p^k", the remainder is always -1.
This equation is significant in mathematics and computer science, as it is used in modular arithmetic to solve problems related to number theory, cryptography, and coding theory.
Yes, this equation can be proven using mathematical induction or other methods in number theory. It is a known theorem and has been proven by many mathematicians.
The conditions for this equation to hold true are that "a" and "p" must be coprime (they have no common factors other than 1) and "p" must be a prime number. "t" and "k" can be any positive integers.
This equation has various real-life applications, including in computer security, where it is used in encryption and decryption algorithms. It is also used in coding theory to detect and correct errors in data transmission.