Is (10^{11}3) a Prime Number? An Exploration into (10^N 3)

Given the mathematical curiosity of understanding whether (10^{11}3) is a prime number, this article delves into the properties of numbers of the form (10^N 3), including theoretical background, practical implementations, and conjectures about the distribution of prime numbers in this sequence.

The Question: Is (10^{11}3) a Prime Number?

It's known that the floor of the square root of (10^{11}3) is 316227, and the number of primes up to 316227 is 27,293. By performing calculations, it is determined that (10^{11}3 100000000003) is a prime number. This result is confirmed by Wolfram Alpha, which identifies 100000000003 as a prime number.

Exploring (10^N 3) for Prime Numbers

While the specific number (boldsymbol{10^{11}3}) is a prime, the more intriguing question is to investigate for what positive integers (N) the number (10^N 3) is prime. This inquiry reveals a rare set of prime numbers.

Using the Miller–Rabin Primality Test

A practical approach to determine if (10^N 3) is prime involves the use of the Miller–Rabin primality test. Below is a Python code example that utilizes this test to check the primality of (10^N 3) for (N) in the range of 1 to 1000:

import random def isPrime(n, k5): if n 2: return False for p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]: if n p 0: return n p s, d 0, n-1 while d % 2 0: s, d s1, d//2 for i in range(k): x pow(random.randint(2, n-1), d, n) if x 1 or x n-1: continue for r in range(1, s): x (x * x) % n if x 1: return False if x n-1: break else: return False return True p [] for i in range(1, 1000): if isPrime(10**i 3): (i) print(p) The output of this code is [1, 2, 5, 6, 11, 17, 18, 39, 56, 101, 105, 107, 123, 413, 426]. This highlights the rarity of such prime numbers and supports the conjecture that the primes of the form (10^N 3) are very sparse.

Natural Density and Upper Bounds

An attempt to refine our understanding involves studying small primes that can divide (10^N 3). For example:

7 divides (10^N 3) when (3^N equiv -3 mod 7), which occurs when (N equiv 4 mod 6). 13 divides (10^N 3) when (N equiv 1 mod 6). Combining these results, if (N equiv 1 mod 3), then (10^N 3) is likely composite.

This analysis suggests that the natural density of the set (S {N in mathbb{N} : 10^N 3 text{ is prime}}) is at most (frac{2}{3}). Further testing with other primes could potentially improve this bound, although it is unlikely to demonstrate that the natural density is 0. It is conjectured that the set (S) could be finite, which would be a stronger result than simply showing that the density is 0.

The exploration of prime numbers of the form (10^N 3) showcases the complexity and intrigue of number theory. Through practical computations and theoretical analysis, we gain insights into the distribution and characteristics of these primes, contributing to a broader understanding of the prime number landscape.