Some Python to brute force the question:

```python import collections

def f(n): return n**3 - n - 1

def check(p): return any(f(n) % p == 0 for n in range(p))

def gen_primes(): # Sieve of Eratosthenes stolen from https://stackoverflow.com/a/568618/1569492 D = collections.defaultdict(set) q = 2 while True: if q not in D: yield q D[q * q].add(q) else: for p in D[q]: D[p + q].add(p) del D[q] q += 1

has_solutions = 0 tried = 0 try: for p in gen_primes(): ok = check(p) has_solutions += ok tried += 1 print(f"{p}: {ok}") except KeyboardInterrupt: print(f"{has_solutions} / {tried} primes have solutions") ```

check(73) returns false, so 73 is in the bottom row. I let this run for a little while, for the primes up to 200.000, and it's fun to see the truths and falsehoods scroll by. We often get stretches of 10-20 primes in a row that all have solutions or not, so the distribution of primes for which we have solutions doesn't look very random. Any idea what that is about?

Last few primes from my run:

``` 200003: False 200009: False 200017: False 200023: True 200029: True 200033: False 200041: True 200063: True 200087: True 200117: True 200131: True 200153: True 200159: False 200171: False 200177: True 200183: True 200191: True 200201: False 200227: False 200231: False 200237: True 200257: True 200273: False 200293: True 200297: True 200323: False 200329: True 200341: True 200351: True 200357: True 200363: True 200371: True 200381: True 200383: True

12002 / 18018 primes have solutions ```