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Prime Factors of Factorial Numbers

Factorial numbers, n!=12nn! = 1 \cdot 2 \cdots n, grow very fast with nn. In fact, n!2πn(n/e)nn! \sim \sqrt{2 \pi n} (n/e)^n according to Stirling's approximation. The prime factors of a factorial number, however, are all relatively small, and the complete factorization of n!n! is quite easy to obtain.

We will make use of the following fundamental theorem:

pabp \mid a b for a prime pp, then pap \mid a or pbp \mid b.

(Here, pap \mid a means that pp divides aa.) This is called Euclid's First Theorem or Euclid's Lemma. For most, it is intuitively clear, but a proof can be found in, e.g., Hardy and Wright: An Introduction to the Theory of Numbers.

An application of this theorem to factorial numbers is that if a prime pp is a divisor of n!n! then pp must be a divisor of at least one of the numbers 1,2,,n1, 2, \ldots, n. This immediately implies

Every prime factor of n!n! is less than or equal to nn.

Conversely, every prime number between 2 and nn must be a prime factor of n!n!.

Let us introduce the notation da(b)d_a(b) as the number of times aa divides into bb. Put more precisely, da(b)=kd_a(b) = k if and only if b/akb/a^k is an integer while b/ak+1b/a^{k+1} is not.

We now seek to determine dp(n!)d_p(n!) for all primes pnp \leq n. From Euclid's First Theorem and the Fundamental Theorem of Arithmetic follows:

dp(n!)=dp(1)+dp(2)++dp(n)d_p(n!) = d_p(1) + d_p(2) + \cdots + d_p(n)

The trick here is not to consider the right-hand side term by term, but rather as a whole. Let us take

42!=140500611775287989854314260624451156993638400000000042! = 1405006117752879898543142606244511569936384000000000

and p=3p=3 as an example. How many of the numbers 1, 2, …, 42 are divisible by 3? Exactly 42/3=14\lfloor 42/3 \rfloor = 14 of them. But this is not the total count, because some of them are divisible by 3 multiple times. So how many are divisible by 323^2? 42/32=4\lfloor 42/3^2 \rfloor = 4 of them. Similarly, 42/33=1\lfloor 42/3^3 \rfloor = 1. And 42/34=42/35==0\lfloor 42/3^4 \rfloor = \lfloor 42/3^5 \rfloor = \ldots = 0. So we have

d3(42!)=14+4+1=19.d_3(42!) = 14+4+1 = 19.

This procedure is easily generalized and we have

(1)
dp(n!)=k=1npk=k=1logp(n)npk.d_p(n!) = \sum_{k=1}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor = \sum_{k=1}^{\lfloor \log_p(n) \rfloor} \left\lfloor \frac{n}{p^k} \right\rfloor.

This identity was found by the french mathematician Adrien-Marie Legendre (see also Aigner and Ziegler: Proofs from The Book, page 8, where it is called Legendre's Theorem).

Doing this for all primes in our example, we get

42!=23931959761131331721922329313741.42! = 2^{39} \cdot 3^{19} \cdot 5^9 \cdot 7^6 \cdot 11^3 \cdot 13^3 \cdot 17^2 \cdot 19^2 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41.

Notice how the exponents do not increase as the prime numbers increase. This is true in general. Assume that pp and qq are both primes and p<qp < q. Then logp(n)logq(n)\log_p(n) \geq \log_q(n) and n/pkn/qkn/p^k \geq n/q^k for all positive integers kk. Using this in equation (1) we get

(2)
dp(n!)dq(n!)  for primes pq with p<qd_p(n!) \geq d_q(n!) \; \text{for primes $p$, $q$ with $p < q$}

and thus

d2(n!)d3(n!)d5(n!)d7(n!)d11(n!)d_2(n!) \geq d_3(n!) \geq d_5(n!) \geq d_7(n!) \geq d_{11}(n!) \geq \ldots

What about dk(n!)d_k(n!) for composite numbers kk? Given the factorization of both n!n! and kk, this is easy to compute. But if, e.g., the multiplicity of all prime factors of kk are the same, then the relation (2) can be used. Consider d10(m)d_{10}(m) for a positive integer mm. Since 10=2510 = 2 \cdot 5 then

d10(m)=min{d2(m),d5(m)}.d_{10}(m) = \min\{ d_2(m), d_5(m) \}.

But if m=n!m=n! then we can use (2) and we have

d10(n!)=d5(n!).d_{10}(n!) = d_5(n!).

For instance,

d10(42!)=d5(42!)=42/5+42/52=8+1=9,d_{10}(42!) = d_5(42!) = \lfloor 42/5 \rfloor + \lfloor 42/5^2 \rfloor = 8 + 1 = 9,

so there are 9 trailing zeros in the decimal representation of 42!.

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