The primes () method gives the list of all primes. Similarly, I need a list (Say PP) that consists of all $p^k-1$ where $p$ is a prime and $k \ge 1$. How to create this infinite list?
Also, I want all possible finite products of elements of PP. Using this I want to understand when two such products are equal.
How to do this? Kindly share your thoughts. Thank you.
Primes() is not a list, but essentially a generator of primes with some additional functions. While it can generate primes one by one, it does not store them all at once anywhere (and so it's not a list).
We can create a generator for integers of the form $p^k - 1$ in their natural order like
PP = (q-1 for q in NN if is_prime_power(q))
As for finite products of elements of PP, again we cannot store them in the list unless they are somehow bounded. So, you need to be a bit more specific how you want to generate those products.
PP = (q-1 for q in NN if is_prime_power(q))
This generator will return the list pf prime any power - 1, without regard to k...
This is what OP asked.
I didn't read it as such ; but you may be right. In which case your brute-force solution is better than what I proposed.
@Max Alekseyev Thank you. How to list the elements less than 100 from PP?
One possible realization as a generator :
def PGm1(k):
"""
Returns a generator for the sequence p(i)^k-1 in i
where p(i) is the i-th prime.
Example : the n first terms of the sequence can be obtained by :
Gk = PGm1(k)
list(Gk.__next__() for u in range(n))
"""
r = 0
while True:
r = r.next_prime()
yield r^k - 1
Example of use :
sage: G2=PGm1(2)
sage: list(G2.__next__() for u in range(5))
[3, 8, 24, 48, 120]
Beware : this generator is infinite. Using it "raw" won't return...
Asked: 2021-04-10 03:40:02 +0100
Seen: 754 times
Last updated: Apr 10 '21
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