Number Theory/Sufficiently/Examples
This page shows many examples, both in mathematics and in practical applications, of the concept of a statement or property being true for sufficiently large / sufficiently late points.
Examples
[edit | edit source]- Example 1: For all dates after February 18, 2017, "Despacito" has more views than "Evolution of Dance." The video "Evolution of Dance" was uploaded to YouTube on April 6, 2006, and on May 19, 2006, it became the most viewed YouTube video with 10 million views. By August 2006, it had risen to 30 million views. While 30 million was very impressive for 2006, it certainly wouldn't make the cut once 2017 rolled around. "Despacito" was uploaded on January 13, 2017, and this video quickly became the viral dance song of 2017. By that point, the video from 2006 had 298 million views, and "Despacito" took only five weeks to surpass it, doing so on February 18, 2017. Six months later, on August 4, 2017, "Despacito" surpassed Wiz Khalifa's "See You Again" to become YouTube's most viewed video, with 2.992 billion views. Later that day, "Despacito" became the first video to reach 3 billion views. The 2006 video is widely treasured as a classic from early YouTube, and many people still revisit it. While that video has stood the test of time and still sees decent viewership, it will never again have more views than "Despacito."
- Example 2: For all dates after May 7, 2018, there is at least one Wayback Machine snapshot for the Mersenne Milestones page between T/2 and T, with 3/24/2018 as the starting date. The Great Internet Mersenne Prime Search is a collaborative project where volunteers from around the world use the Prime95 software to run primality tests for Mersenne numbers. As the project has discovered 18 prime numbers since 1996, and most of them were the "largest known prime" at the time of their discovery, the project's website banner reads, "Finding World Record Primes Since 1996." The "Milestones" page on the website shows the progress towards upcoming near-future milestones. As with other popular, constantly changing pages, this page is frequently crawled by the Wayback Machine and other crawlers such as Archive.today. Prior to 2016, this was not the case; in 2015 and earlier, this page was rarely crawled. The page was crawled by the Wayback Machine on November 4, 2010; this didn't happen again until February 9, 2013. The next capture was a year and a half later, on September 11, 2014. Starting on January 21, 2016, this page started getting crawled nearly every day, with several "exception" periods. There was a 9-day gap in the early autumn of 2017, where 9/23/2017 was followed by 10/02/2017, then in 2018, there were two 23-day gaps, between 4/14/2018 and 5/07/2018, then, eight months later, between 12/12/2018 and 1/04/2019. During the 23 days between April 14 and May 7, the age of a video dated 3/24/2018 increased from 21 days old to 44 days old, which is more than doubling. This will never happen again. And even with "Save Page Now" temporarily inaccessible due to the DDOS attacks on the Internet Archive this month, I have confidence the feature will be restored. People on Reddit have been calling others out for making posts that "complain" about the Internet Archive "still" not being fully restored, telling them that this is a security project that will take some time, and that those who are complaining should exercise some patience. Let me tell you this; now that 3/24/2018 was six and a half years ago, this shutdown would have to extend all the way to the spring of 2031 in order for the age of 3/24/2018 to double during this shutdown. Once this shutdown ends, which will undoubtedly happen in the coming weeks or even days, crawls on this page can and will resume.
- Example 3: The age of "Bad Romance" will never again "double" during a gap between Wayback Machine captures of the Mersenne Milestones page. In other words, if we take November 24, 2009, the release date of "Bad Romance" as the starting date, then from February 9, 2013, onwards, there will always be at least one Wayback Machine capture between T/2 and T, where T is the current date, where T/2 is the date on which "Bad Romance" was half as old. There were zero Wayback Machine captures for the Mersenne Milestones page between 11/04/2010 and 2/09/2013. On February 9, 2013, at 01:48:01 UTC, the page was crawled for the first time in two years and three months. Therefore, February 8, 2013, the previous day, was (and always will be) the last date for which, using 11/24/2009 as the starting date, there were no snapshots of the Mersenne Milestones page in the "second half." On February 8, 2013, "Bad Romance" was 1,172 days old. Half of 1,172 is 586, and "Bad Romance" was 586 days old on July 3, 2011. No captures were made between 7/03/2011 and 2/08/2013. However, for all later dates (Feb. 9, 2013, as well as all later dates), there are captures in the second half. Even the extended disruption and an extended "Temporarily Unavailable" period for "Save Page Now" due to a DDOS attack will not result in another "age doubling," as "Bad Romance" is nearing its 15th anniversary this November.
- Example 4: After January 17, 2021, there is at least one Miley Cyrus song that has appeared on Todd in the Shadows' "best hit songs" list. "Midnight Sky" took #4 on Todd's "best hit songs of 2020" list.
- Example 5: After May 6, 2013, Michelle Knight, Amanda Berry, and Gina DeJesus are not stuck in Ariel Castro's house. They are free!
- Example 6: After December 30, 2023, my "Autumn 2023 loud cough ordeal" is over. The cough started on November 19, 2023, and it persisted for 40 days before it finally subsided on December 29.
- Example 7: After December 28, 2023, the statement, "It is 7:00 p.m. Eastern Time during the Autumn 2023 loud cough ordeal" will never be true again. Although, clearly, 7:00 p.m. Eastern Time happens every 24 hours, the entire statement is transient and can never be true again because the "Autumn 2023 loud cough ordeal" is over and will never again not be over. Since the cough FINALLY went away the evening of December 29, 2023, at around 5 or 6 in the evening, the last time this statement was (unfortunately) true was December 28, 2023, at 7:00 p.m. EST. This statement was true 40 times in 40 days, but it will never be true again.
- Example 8: Starting December 19, 2016, the best (lowest) Wii Fit Age I can get is 20. I was born on December 19, 1996, so 12/19/2016 was my twentieth birthday. In all titles in the Wii Fit series, a lower Wii Fit Age indicates better health and better fitness, while a high Wii Fit Age indicates you are weaker and less healthy than you should be. The minimum Wii Fit Age can be calculated with the following formula: min(20, x) where x is your current age. If you are 19 or younger, the minimum is your current age. If you are 20 or older, it is set to 20 instead. From 12/19/2016 onwards, the lowest Wii Fit Age I can get is 20, but before this date, it was possible to get a lower Wii Fit Age. In the summer of 2009, when I played Wii Fit for the first time, I used to play alongside my brother. I was 12 at the time, and he was nine. When I did superbly, I would get a Wii Fit Age of 12, and when it was Bryan's turn to play, he would do superbly and get a 9. In late 2009, my Wii Fit disc stopped working, and it would be five years before I'd get to play Wii Fit again. On November 29, 2014, the weekend after Thanksgiving, that finally happened. I was 17, about to turn 18 in December, and Bryan was 14, about to be 15 in January. In the summer of 2015, I played Wii Fit frequently, and while I would sometimes get a high Wii Fit Age, other times, I performed well and got an 18. Seeing the Wii Balance Board (on the television screen) say the words, "Your Wii Fit Age is exactly the same as your age! You're still very young!" felt so good.
- In December 2017, in the days shortly after my 21st birthday, I went to take the balance tests, and when I passed them, I was surprised the game was giving me a 20 instead of a 21. My first impression was: Maybe Wii Fit doesn't like the number 21 for some reason. Then, as I turned 22, 23, 24, and so on, the minimum stayed at 20.
- Example 9: From September 25, 2013, onwards, the lifetime "daily views/day" average for "Wrecking Ball" is less than 10 million. This video was uploaded on September 9, 2013, at noon EDT (12:00:38 to be exact). Within its first 24 hours, the video got 19,300,000 views. On September 15, the video broke the record for being the fastest video to reach 100 million views, doing so in only six days. That's an average of 16.6 million views/day. By September 25, 2013, the daily average fell to under 10 million views/day, with 152 million views in 16 days (9.5 million views/day). Now that the video is 11 years old, over 4,000 days old, it will never again rise above a lifetime daily average of 10 million views per day: achieving that feat today would have required it to have over 40 billion views, nearly triple the number of views on "Baby Shark"! By January 1, 2014, the video had 453 million views in 114 days, bringing its daily average to just under 4 million views per day.
- Example 10: From September 1, 2012, onwards, YouTube servers have been tracking the "view duration" for all YouTube videos.
- Example 11: After the 2018 Atlantic hurricane season, "Florence" cannot be used as the name of an Atlantic storm. The name "Florence" was given to ten storms in the Atlantic basin between 1953 and 2018, making it one of, if not the, most common names for hurricanes and tropical storms. See List of storms named Florence at Wikipedia for a list of Atlantic storms with this name. This list will forever be complete. There will never be an eleventh time. In 2018, Hurricane Florence resulted in 54 deaths and $24.2 billion in damages. Due to the extensive damage and loss of life, the name "Florence" was retired and will not be used again for any future Atlantic storms. It was replaced with "Francine" for the 2024 season.
- Example 12: After January 1, 2019, babies born in the second millennium are 18 or older.