Are there any day to day situations in which number theory could be applied to? For example, I was trying to make a mosaic and I realized that diophantine equations could be really useful if I needed to find how many pieces of certain dimensions needed to be used. Are there any other real life situations in which number theory would be useful?
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$\begingroup$The RSA crypto-algorithm, which critically depends on number theory. That's billions of dollars in global revenue every day, which should hopefully qualify as a real life application.
$\endgroup$ 1 $\begingroup$Cryptography, hash functions, and random number generators. The general number field sieve obviously also uses Number Theory heavily. The same applies to Shor's Algorithm. Although at a superficial glance number theory may look to have no 'real life' applications, its principles and ideas are widely used in computer science and especially coding.
“To a mathematician, real life is a special case.”
$\endgroup$ $\begingroup$The most important application of number theory is that it is the key foundation of cryptography. Our strong encryption algorithms and systems have developed because of the impetus provided by number theory. For example, your data cannot be easily accessed by anyone because of the strong encryption system. Moreover number theory is useful in the study of binary codes and other related concepts. This is the main real life applications of number theory. Other than those, there are myriad of applications which are useful for applied mathematicians and physicists. For example, the q series is extremely useful in the study of strings. Then supersymmetric functions like the mock theta functions and theta functions are used for a large number of advanced purposes. Over-all number theory is extremely useful for applied mathematicians and physicists. But to understand those applications you must have a sound knowledge in both.
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