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Why are prime numbers important in real life? What practical use are prime numbers?
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Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.
For a mathematician it certainly is!Definitely a real life application.
Ask periodical cidacas why they use prime numbers for the cycle length.
The fact that the surviving periodical cicadas have life cycles built on prime numbers may have conferred key survival advantages. A prime-numbered lifespan means that predators cannot match their own shorter life cycles to the availability of cicada prey. For instance, if the cicadas had even-numbered lifespans, a predator with a two-year life cycle could expect a cicada feast, and a subsequent population bump, every few generations, because all even numbers are divisible by two. As explained in 2001 by a trio of researchers from the University of Chile and the Max Planck Institute of Molecular Physiology in Germany, “a prey with a 12-year cycle will meet — every time it appears — properly synchronized predators appearing every 1, 2, 3, 4, 6 or 12 years, whereas a mutant with a 13-year period has the advantage of being subject to fewer predators.”
from Wikipedia:
Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors.
Poor Hardy! He went to his grave proud of the uselessness of number theory. Strange that a discipline studied for millennia suddenly became useful only a couple of decades ago.