Prime Gaps & Twin Primes

Prime Gaps

A prime gap is the difference between two consecutive primes $p$ and $q$ where $q$ is the next prime after $p$:

$g(p) = \text{next\_prime}(p) - p$

The smallest possible gap is 2 (since consecutive integers differ by 1, and at least one must be even, so consecutive primes must differ by at least 2 — except for the pair (2, 3)).

Twin Primes

Two primes $(p, p+2)$ are called twin primes — the smallest possible gap between odd primes. Examples: $(3,5)$, $(5,7)$, $(11,13)$, $(17,19)$, $(29,31)$, …

The Twin Prime Conjecture (unsolved!) states there are infinitely many twin prime pairs.

def is_prime(n):
    if n < 2: return False
    if n == 2: return True
    if n % 2 == 0: return False
    for i in range(3, int(n**0.5) + 1, 2):
        if n % i == 0: return False
    return True

def next_prime(n):
    candidate = n + 1
    while not is_prime(candidate):
        candidate += 1
    return candidate

def prime_gap(p):
    return next_prime(p) - p

def is_twin_prime(p):
    return is_prime(p) and prime_gap(p) == 2

print(prime_gap(11))      # 2  (next prime is 13)
print(is_twin_prime(11))  # True
print(prime_gap(7))       # 4  (next prime is 11)
print(is_twin_prime(7))   # False

Maximal Gaps

Prime gaps grow on average (by the Prime Number Theorem), but they are irregular. After $p = 23$, the next prime is $29$ — a gap of 6. After $p = 89$, the next is $97$ — a gap of 8.

Polignac's Conjecture

Every even number $n$ occurs as a prime gap infinitely often. This is unsolved for every specific even value, including 2 (the Twin Prime Conjecture).

Your Task

Using the is_prime and next_prime helpers provided in the starter code, implement:

• prime_gap(p): gap from $p$ to the next prime
• is_twin_prime(p): whether $p$ is the smaller of a twin prime pair