A number greater than one billion, with deep mathematical meaning.

Question

The most famous large number that have appeared in mathematical papers is the Graham number. However, the Graham number is an upper bound, and its largeness has no meaning.

I would like to know about huge numbers that have deep mathematical meaning.

What are some mathematically significant numbers that are larger than a billion?

For example, the order of the monster group.

UPDATE:

• Since the Mersenne primes are an important but prime set, it seems obvious that large ones exist. I want you to name a single number.

• Numbers that have no eye-catching features other than size are not included.

• Artificial numbers such as "pi times a billion" are excluded.

• The Fermat number $F_5$ is certainly important, but I am looking for numbers that has important points outside of its historical context.

Answer 1

Littlewood proved that, contrary to then popular belief, there were infinitely many integers $n$ such that there were more primes less than $n$ congruent to $3 \pmod{4}$ than to $1$.

The first such $n$ (which I do not know, but which is probably known) would be an answer to this question.

This abstract of Carter Bays and Richard H. Hudson's On the Fluctuations of Littlewood for Primes of the Form $4n \pm 1$ suggests some other candidates.

Let $\pi_{b,c}(x)$ denote the number of primes $\leqslant x$ which are $\equiv c (\operatorname{mod} b)$. Among the first 950,000,000 integers there are only a few thousand integers $n$ with $\pi_{4,3}(n) > < \pi_{4,1}(n)$. The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for $L(s)$. The discovery that the majority of all integers $n$ less than $2 \times 10^{10}$ with $\pi_{4,3}(n) < > \pi_{4,1}(n)$ are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.

On the same topic, from the same paper, reported in Wolfram

Similarly, consider the list of the first n primes   {p_3,p_4,...,p_n} (mod 3), 
skipping p_1=2 and p_2=3 since 3=0 (mod 3).  This list contains equal numbers of 
remainders 2 and 1 at the values n=4, 6, 8, 12, 14, 22, 38, 48, 50, ... (OEIS A096629). 
The first value of n for which the list is biased towards 1 is n=23338590792, as first found by 
Bays and Hudson in 1978 (Derbyshire 2004, p. 126), giving the first few such values as 
23338590792, 23338590794, 23338590795, 23338590796, ... (OEIS A096630). 

Answer 2

A few examples:

Magic Squares

$46656000000$: the smallest known, and likely smallest possible, magic product for a sixth-order multiplicative pandiagonal magic square. Sixth order is the smallest for which this problem is not definitively solved.

Digital combinations

$13223140496$: when this base 10 representation is concatenated with itself, the result becomes a square. This is the smallest known such example beginning with a nonzero digit.

$105263157894763842$: the smallest number whose double is a one-step cyclic permutation of itself; from the properties of the full-repetend decimal representation of $1/19$.

$666...666$ (61 digits): This number consists entirely of one digit in base 10, but it has a factor in which all ten digits appear equally (six times apiece), obtained by dividing the given number by $61$; from the full-repetend and equal-ocurrence properties of the decimal representation of $1/61$. Using $7,8$ or $9$ in place of $6$ gives similar results, but a smaller digit fails because a zero is lost at the beginning of the quotient.

Answer 3

Large primes are important, and types of primes are important as: Twin primes, Mersenne primes and many types.

These primes are important in number theory.

Do they give answer for you?

See the page for seeing the list of Mersenne primes:https://www.mersenne.org/

See the following nice number, I learned it from page $116$ of Elementary Number Theory With Applications By Thomas Koshy:

For the curious minded, the largest known prime, all of whose digits are also prime, is $72323252323272325252 * \frac{10^{3120} − 1}{10^{20} − 1} $ . Discovered in 1992 by Harvey Dubner of New Jersey, it has 3120 digits.

Another interesting number is:

$2^{70}=1,180,591,620,717,411,303,424$.

The sum of its digits is $70$.