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Suppose by contradiction, there are finite prime numbers. Let us denote these numbers as p1, p2, p3... pk, the last prime number.
Let M = p1 + p2 + p3 + ... + pk + 1
M is not a prime number because M is greater than any of the prime numbers. Any whole number greater than 1 is divisible by a prime number.
If we subtract all the prime numbers from M, we get:
M - p1 - p2 - p3 - ... - pk = 1.
By this logic, one should be divisible by a prime number bigger than it and give a whole number. But we know that is not the case because that would be retarded. Therefore there are infinite prime numbers
⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠄⠄⠄⠄
⠄⠄⠄⠄⠄⠄⠄⠄⠄⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠄⠄⠄
⡄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠉⠄⠉⠉⠉⣋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢷⡀⠄⠄
⣿⡄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠠⣾⣿⣷⣄⣀⣀⣀⣠⣄⣢⣤⣤⣾⣿⡀⠄
⣿⠃⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⣹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢁⣠
⣿⣿⣄⣀⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠉⠉⣉⣉⣰⣿⣿⣿⣿⣷⣥⡀⠉⢁⡥⠈
⣿⣿⣿⢹⣇⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠒⠛⠛⠋⠉⠉⠛⢻⣿⣿⣷⢀⡭⣤⠄
⣿⣿⣿⡼⣿⠷⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⢀⣀⣠⣿⣟⢷⢾⣊⠄⠄
⠉⠉⠁⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠈⣈⣉⣭⣽⡿⠟⢉⢴⣿⡇⣺⣿⣷
⠀⠀⢰⠀⠀⢂⠡⠼⠊⠁⠈⠉⠉⠁⠐⢢⡀⠀
⠀⠀⠘⡤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠟⠀
⠀⠀⠀⡔⠤⡀⠀⠀⠰⣶⠖⢰⠂⠀⠉⠀⠀⠀
⠀⠀⡰⠉⠐⠠⠌⠉⠁⠠⡕⠺⡄⠀⠀⠀⠀⠀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⠼⠀⢸⠀⠀⠀⠀⠀
⠀⠘⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀
⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⢰⠀⠀⠀⠀
⠀⠘⡦⠔⠤⠃⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀
⢠⣔⣉⠆⠀⠀⠀⠀⢄⠀⠀⠀⣇⠀⠀⠀⠀⠀
⠀⠀⠀⢰⠀⠀⠀⠀⢸⠆⠀⠀⢸⠀⠀⠀⠀⠀
⠀⠀⠀⡆⠀⠀⠀⢀⠈⡢⠀⠀⢀⢕⠀⠀⠀⠀
⠀⠀⠀⠑⠢⠄⣀⣀⣱⠸⠂⠀⠚⠉⠀⠀