Counting Prime Numbers Less Than 1,000,000 with the Last Digit 3

The fascinating world of prime numbers never ceases to amaze! When considering prime numbers less than 1,000,000, approximately 25% of them end in the digit 3. In other words, out of all the primes up to a million, a quarter of them have that distinctive last digit.

Identifying the Prime Numbers

To find the number of prime numbers less than 1,000,000 that end with the digit 3, we can use the following reasoning:

Identify the Range

We are looking for prime numbers that are less than 1,000,000. This is a well-defined upper limit for our search.

Consider the Last Digit

A prime number greater than 3 cannot be even and hence cannot end with 0, 2, 4, 6, or 8. Additionally, it cannot end with 5 except for the number 5 itself. Therefore, the possible last digits for prime numbers are 1, 3, 7, and 9.

Count the Primes

According to the prime number counting function and known results, there are approximately 78,498 prime numbers less than 1,000,000. To find those ending in 3, we can check the distribution of the last digits of these primes.

Distribution of Last Digits

The last digit of prime numbers tends to be uniformly distributed among 1, 3, 7, and 9. Given that there are four options, we can estimate that each last digit would account for about 25% of the total number of primes.

Calculation

Estimating the number of primes ending in 3:

Estimated number of primes ending in 3 78,498 / 4 ≈ 19,624.5

Since we cannot have a fraction of a prime number, we round this to the nearest whole number. Therefore, the estimated number of prime numbers less than 1,000,000 that end with the digit 3 is approximately 19,624.

Methods for Exact Counts

For an exact count, a sieve method or computational approach would be required. However, this estimation provides a good approximation based on the uniform distribution of the last digits of primes.

Additional Insights

According to the Prime Numbers Wiki, there are 106 three-digit prime numbers between 101 and 1,000. To extend this to all primes less than 1,000,000, we can use WolframAlpha to compute the exact count.

Conclusion: So there are 78,498 prime numbers below one million. Among them, approximately 25% end in the digit 3!