The AKS algorithm for testing whether a number is prime is a polynomial-time algorithm based on an elementary theorem about Pascal triangles.

The theorem on which the test is based can be stated as follows:

   a number   



p


{\displaystyle p}

   is prime   if and only if   all the coefficients of the polynomial expansion of
 



(
x
−
1

)

p


−
(

x

p


−
1
)


{\displaystyle (x-1)^{p}-(x^{p}-1)}


are divisible by   



p


{\displaystyle p}

.



Example

Using   



p
=
3


{\displaystyle p=3}

:

         (x-1)^3 - (x^3 - 1)
            = (x^3 - 3x^2 + 3x - 1) - (x^3 - 1)
            = -3x^2 + 3x



And all the coefficients are divisible by 3,   so 3 is prime.





 Note:
This task is not the AKS primality test.   It is an inefficient exponential time algorithm discovered in the late 1600s and used as an introductory lemma in the AKS derivation.



Task



 Create a function/subroutine/method that given   



p


{\displaystyle p}

   generates the coefficients of the expanded polynomial representation of   



(
x
−
1

)

p




{\displaystyle (x-1)^{p}}

.
 Use the function to show here the polynomial expansions of   



(
x
−
1

)

p




{\displaystyle (x-1)^{p}}

   for   



p


{\displaystyle p}

   in the range   0   to at least   7,   inclusive.
 Use the previous function in creating another function that when given   



p


{\displaystyle p}

   returns whether   



p


{\displaystyle p}

   is prime using the theorem.
 Use your test to generate a list of all primes under   35.
 As a stretch goal,   generate all primes under   50   (needs integers larger than 31-bit).


References

 Agrawal-Kayal-Saxena (AKS) primality test (Wikipedia) 
 Fool-Proof Test for Primes - Numberphile (Video). The accuracy of this video is disputed -- at best it is an oversimplification.


AKS<-function(p){ 
  i<-2:p-1 
  l<-unique(factorial(p) / (factorial(p-i) * factorial(i))) 
  if(all(l%%p==0)){   
    print(noquote("It is prime."))   
  }else{   
   print(noquote("It isn't prime."))
  }
}