Question
Salem–Spencer sets cannot contain any non-trivial examples of these sequences, which are the subject of Szemerédi's theorem. For 10 points each:
[10e] Name these sequences in which the difference between any two consecutive terms is constant, unlike in geometric sequences.
ANSWER: arithmetic sequences [or arithmetic progressions]
[10h] In 2004, Szemerédi's theorem was extended with this result, which states that the sequence of prime numbers contains arbitrarily long arithmetic sequences.
ANSWER: Green–Tao theorem
[10m] The Green–Tao theorem led to improved results concerning the first Hardy–Littlewood conjecture, which generalizes this conjecture. This conjecture has been verified up to 10-to-the-388342, somewhat larger than the 10-to-the-18 upper bound for Goldbach’s conjecture.
ANSWER: twin prime conjecture
<Other Science>
Summary
| 2024 ACF Winter at UC Berkeley | 2024-11-16 | Y | 3 | 20.00 | 100% | 33% | 67% |
| 2024 ACF Winter at Clemson | 2024-11-16 | Y | 7 | 14.29 | 100% | 14% | 29% |
| 2024 ACF Winter at Lehigh | 2024-11-16 | Y | 7 | 15.71 | 100% | 14% | 43% |
| 2024 ACF Winter at Northwestern | 2024-11-16 | Y | 9 | 17.78 | 100% | 33% | 44% |
| 2024 ACF Winter at Ohio State | 2024-11-16 | Y | 6 | 13.33 | 100% | 17% | 17% |
| 2024 ACF Winter at Online | 2024-11-16 | Y | 7 | 12.86 | 100% | 29% | 0% |
| 2024 ACF Winter at UBC | 2024-11-16 | Y | 2 | 25.00 | 100% | 100% | 50% |
| 2024 ACF Winter at Central Florida | 2024-11-16 | Y | 5 | 12.00 | 100% | 0% | 20% |
| 2024 ACF Winter at Oxford | 2024-11-16 | Y | 10 | 14.00 | 100% | 30% | 10% |
Data
| Indiana B | WashU D | 10 | 0 | 0 | 10 |
| Purdue A | Miami | 10 | 0 | 0 | 10 |
| WashU B | Northwestern A | 10 | 10 | 10 | 30 |
| Notre Dame | UChicago C | 10 | 0 | 0 | 10 |
| Indiana A | Purdue C | 10 | 10 | 0 | 20 |
| UChicago B | Purdue D | 10 | 0 | 10 | 20 |
| UIUC A | SIUE A | 10 | 10 | 0 | 20 |
| UIUC C | UChicago D | 10 | 0 | 10 | 20 |
| UChicago A | WashU C | 10 | 10 | 0 | 20 |