Number Theory/Sufficiently

In mathematics and everyday life, a statement is said to be eventually true if it is always true beyond a certain constant, finite point. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers" and can be also extended to the class of properties that apply to elements of any ordered set. For instance, since all prime numbers except two are odd, we can write, "All sufficiently large primes are odd" or "Eventually, all primes are odd."

Motivation and definition

For an infinite sequence, mathematicians are often more interested in the long-term behaviors of the sequence than the behaviors it exhibits early on. In which case, one way to formally capture this concept is to say that the sequence possesses a certain property eventually, or equivalently, that the property is satisfied by one of its subsequences $(a_{n})_{n\geq N}$, for some $N\in \mathbb {N}$.

The concept of "sufficiently large" plays an important role in the concept of limits. The definition of a sequence of real numbers $(a_{n})$ converging to some limit $a$ is:

For each positive number $\varepsilon$, there exists a natural number $N$ such that for all $n>N$, $\left\vert a_{n}-a\right\vert <\varepsilon$.

When the term "eventually" is used as a shorthand for "there exists a natural number $N$ such that for all $n>N$", the convergence definition can be restated more simply as:

For each positive number $\varepsilon >0$, eventually $\left\vert a_{n}-a\right\vert <\varepsilon$.

This does not necessarily mean that any particular value for $a$ is known, but only that such an upper bound exists. The phrase "sufficiently large" should not be confused with the phrases "arbitrarily large" or "infinitely large". For example, there exist arbitrarily large powers of two (2, 4, 8, 16, 32, ...), but it is certainly not true that all sufficiently large integers are powers of two. "Sufficiently large" is a special case of the notion of "almost all," which generally allows for infinitely many exceptions. For example, almost all positive integers are composite, but it is not true that "all sufficiently large integers are composite" because there are infinitely many primes.

Examples

• Eventually, all primes are congruent to ±1 modulo 6. (This is true for any prime greater than or equal to 5.)
• In base 10 (decimal), the factorial of any large enough integer is divisible by 100. (This is true for any integer greater than or equal to 10.)
• The square of a prime is eventually congruent to 1 mod 24. (This is true for any prime number greater than or equal to 5.)

Timeline theory

In timeline theory, a statement is said to be true for all sufficiently late dates if it is always true beyond a constant, finite point in time. Some statements were true in the past but will never be true again. Once you turn 20 years old, your ideal "Wii Fit Age" in any Wii Fit game is 20. Whether you're 21, 25, or 80, you can always shoot for a Wii Fit Age of 20 if you perform superbly on the Body Test. If you are 19 or younger, the minimum Wii Fit Age is your current age, but for all sufficiently late dates (from your 20th birthday onwards), the minimum takes a constant value of 20.

It is not always easy to tell whether an observed event can happen indefinitely or will eventually become impossible. When you are in high school, you may see your current teacher nearly every day. But it is possible that, due to unexpected or unforeseen circumstances, the teacher leaves without warning. If you liked this teacher, especially if the teacher was a friend to you, you may be taking the news hard. After the teacher leaves, you may see the teacher a lot less frequently, or in many cases, not at all.

Even so, there are some situations where we can be sure that an event will not occur again. Define a "transient" event as an event that happens only finitely many times, then define a "recurrent" event as one that would happen infinitely many times, assuming an infinite timeline. Also, an event is said to be discrete if, between any two instances of the event, the gap in time between them is neither zero nor infinitely short. Thus, for a discrete event, any sufficiently short period of time includes at most one instance of the event.

Interval Theorem

Let S be any finite and fixed interval of time with fixed endpoints. An interval is defined as starting from one moment in time and ending at another. If X is a repeatable, discrete event that occurs at least once during interval S, then there exist only finitely many instances of X during S.

Therefore, even if X is recurrent, XS (denoting the event of X happening during S) is transient.

• Example 1: In any given calendar year, there are only finitely many Sundays, 52 or 53, depending on the year. In 2012, there were 53 Sundays, the last of which was December 30, 2012.
• Example 2: On November 19, 2023, while at the Museum of Science, I got a terrible sickness and cough that persisted for six weeks. The cough finally subsided on December 30, 2023, after 40 days, after six long, frustrating weeks. During those 40 days, I met with my tutor EK Lung six times, since we meet once a week. The meeting occurs on Sundays, at 7 p.m., so the last meeting with him during my ordeal was December 24, 2023, at 7:00 p.m.
• Example 3: Using the same 40-day ordeal that I used in the previous example: Every day, at 7 p.m. EST, I go to the YouTube page for Miley Cyrus' "Flowers" and collect the view count of the video. Since my cough lasted 40 days, I did the "7:00 routine" 40 times during the cough ordeal. The last time was December 28, 2023, at 7 p.m. Since this cough ordeal has ended, and since it will never again not be over, the statement, "I am currently doing the 7:00 routine during my Nov/Dec 2023 loud cough ordeal," will never be true again. As I am writing this article on October 26, 2024, this statement has now been false for 303 days. The negation of the statement ("I am not currently doing ...") has been true for 303 days in a row.
• Example 4: Justin Bieber's video for "Baby" used to be YouTube's most-viewed video. This video held that title for 862 days, from July 16, 2010, until November 24, 2012. At the League School of Greater Boston, the last day of school during that 862-day interval was Wednesday, November 21, 2012, the day before Thanksgiving.
• Example 5: