Question
The Auslander–Buchsbaum theorem gives that all regular local rings are UFDs, meaning that each of their elements can be uniquely decomposed into these numbers. For 10 points each:
[10e] Name these numbers. The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely represented as the product of these numbers.
ANSWER: primes [or prime numbers]
[10h] Uppercase and lowercase instances of this Greek letter denote functions that count the prime factors and the unique prime factors of a number, respectively. This letter with subscript 1 denotes the first uncountable ordinal.
ANSWER: omega
[10m] The prime omega functions can be used to calculate the Möbius function, which evaluates to zero on numbers with a prime factor of multiplicity at least this number. Mersenne primes are of the form “[this number] to the n minus 1.”
ANSWER: two
<Science - Other Science>
Summary
| 2023 ACF Fall (Rutgers) | 10/14/2023 | Y | 5 | 18.00 | 80% | 80% | 20% |
Data
| Lehigh | Columbia A | 10 | 0 | 0 | 10 |
| Penn | NYU B | 10 | 0 | 10 | 20 |
| Princeton | NYU C | 10 | 10 | 10 | 30 |
| Swarthmore A | University of Delaware B | 10 | 0 | 10 | 20 |
| Maryland | Swarthmore B | 0 | 0 | 10 | 10 |