1. What is Sum of Primes?

The sum of primes up to a given number n is the total of all prime numbers less than or equal to n. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

Where:

• \( p \) — Prime numbers
• \( n \) — Upper limit integer
• \( \sum \) — Summation symbol

Explanation: The calculator iterates through all numbers from 2 to n, identifies prime numbers using trial division, and sums them up to provide the final result.

3. Importance of Prime Numbers

Details: Prime numbers are fundamental in number theory and have practical applications in cryptography, computer science, and mathematics. The distribution and properties of primes have been studied for centuries.

4. Using the Calculator

Tips: Enter any integer n between 2 and 1,000,000. The calculator will sum all prime numbers from 2 up to and including n (if n is prime).

5. Frequently Asked Questions (FAQ)

Q1: What is the largest number I can input?
A: The calculator accepts numbers up to 1,000,000 for reasonable computation time.

Q2: Are there any known formulas for sum of primes?
A: While there are asymptotic formulas (like the prime number theorem), there's no simple closed-form formula for the exact sum of primes up to n.

Q3: What is the sum of first 10 prime numbers?
A: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129

Q4: Why is 1 not considered a prime number?
A: By definition, primes must have exactly two distinct positive divisors. 1 has only one divisor (itself), so it's not prime.

Q5: What are some applications of prime sums?
A: Used in mathematical research, cryptography algorithms, and testing computational efficiency of prime-finding algorithms.