First for it's importance in the field of abstract algebra:

  • This function returns the cardinality (order) of the group $U(n)$ closed under modular multiplication.

  • This function also returns the upper bound for the order of an arbitrary element in the $U(n)$.

And after that it's computational importance in modular arithmetic:

  • Since residue exponention is not well-defined, reducing the exponent modulo $\varphi(m)$ is our only way out for simplyfing modular exponents due to the Euler-Totient theorem.

  • With the same way above, if exponent of the element is relatively prime with the $\varphi(m)$, we can compute modular inverse of the exponent and with the help of it we can calculate modular roots. It is essential to RSA.

Are my points true? What i can add to this list?

  • 4
    $\begingroup$ For the second point, see Carmichael's function, which gives the best upper bound for multiplicative orders modulo $n$. $\endgroup$ – lhf Jul 5 at 14:06
  • $\begingroup$ I'd say the core of modular arithmetic is the decomposition of the groups $\Bbb{Z}_n,\Bbb{Z}_n^\times$ and the ring $\Bbb{Z}_n$ in term of prime powers $p^k | n$ and $\varphi(n)$ encodes one of the main parameters of those things $\endgroup$ – reuns Jul 5 at 22:03
  • $\begingroup$ This question overlaps math.stackexchange.com/q/2816053. $\endgroup$ – Steven Clark Jul 6 at 17:41

A few things:

  • You reduce many term polynomials, to just a few terms mod any value. (see here as applied to an integer, a less general form of polynomial).

  • It shreds power towers down to size, with repeated use.

  • It allows us to work in smaller numbers, rather than potentially trillion digit numbers.

  • It allows us to pigeonhole principle coprime variable sets.

  • You can generalize it to products of coprime arithmetic progressions. The product of the first 4 numbers in arithmetic progression 10k+9 have a product that is 1 mod 10 for example (193,401 for those wondering).

  • Can be used to limit long division in finding the reptend length of fractions with coprime denominator in a given base.

  • Cryptography ( forgot this)

    • etc.

Another reason from abstract algebra:

If $n$ is a positive integer then every group of order $n$ is cyclic if and only if $\gcd(n,\varphi(n))=1$.

You can read about the history and proof of this result here.


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