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Euler function  ?: N --> N defined as:   ?(n) = {quantity of values 1 ≤ k < n with m

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Euler function  ?: N --> N defined as: 

 ?(n) = {quantity of values 1 ≤ k < n with m.c.d(k,n) =1

 

 ?(7) = 6 because k = 1,2,3,4,5,6 are values between 1 and 6 coprimes with 7

 

Knowing the next properties are satisfied:

?(p) = (p-1) for each p prime and if m ?N  ?: (p^m) = p^m (1-1/p)

If m.c.d(p,q) = 1, then ? (p,q) = ?(p) . ?(q)

if m.c.d (a,p) = 1, then a?^p = 1mod(p)

 

What i want to solve to practice is:

Find values of ?(4),  ?(13),  ?(2500),

Find values of n ?N where   ?(n) is equal to 1,2 and 3

Find  ?(64719), and with that answer calculate 53³⁹⁷⁴⁸mod(64719)