406.sagews

First step. Goal: to know using Sage if a number is an integer .
The idea is to introduce the set on integers ( let us name is "N" )

N=IntegerRing

3.7 in N()

False

4 in N()

True

Now we can easily ask Sage if (n^x-n) is multiple of x ( here with n=12 and x=13 )

n=12;x=13

(n^x-n)/x in N()

True

Second step. Goal : to know using Sage if a number x is prime.
The idea is to use the fuction " x in Primes()"

3 in Primes()

True

How to test if (n^x-n) is multiple of a given x for some different values n ?
The idea is to use pour x the set of integers n ( between two values ) such that (n^x-n)/x  is integer
for example: for x=8

x=8;w = [n for n in range(1,10) if (n^x-n)/x in N()]

w

[1, 8, 9]

Set(w).cardinality()

3

That means for x=13 we have only 3 numbers between 1 and 10 having the property (n^x-n) divisible by x

By this way you can easily find a criteria for the property of divisibility for all integer n between 1 and some "max" value...

def divisib(x,max): return Set([n for n in range(1,max) if (n^x-n)/x in N()]).cardinality()==max-1

divisib(14,10)

False

divisib(17,10)

True

exeptions=Set([k for k in range(1,20) if divisib(k+0,20) and k not in Primes() ])

exeptions

{1}

This last result means : for x between 1 and 20 we have only one number (x=1) not prime having the property... Now you can see if they are some more exeptions ...