Is 677 a Prime Number? Yes!

Number: 677

✅ Yes! It's a Prime Number

Prime numbers are only divisible by 1 and itself.

⬅ Previous Prime: 673 Next Prime: 683 ➡

🧠 Did You Know?

2 is the smallest and only even prime number.
Prime numbers play a key role in cryptography.
Every number is either prime, composite, or 1.
There are infinitely many prime numbers.
Mathematicians use prime numbers in advanced theories.

🎯 Prime Number Database

Explore comprehensive data about prime numbers and their fascinating properties.

🔢 First 100 Prime Numbers

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

Click any prime to analyze it!

📍 25 Prime Numbers Near 677

607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761

📈 Prime Gaps Analysis Around 677

Twin Primes (Gap = 2)

617–619, 641–643, 659–661 ...

Cousin Primes (Gap = 4)

613–617, 643–647, 673–677, 739–743, 757–761 ...

Gap-6 Primes

607–613, 647–653, 653–659, 677–683, 727–733, 733–739, 751–757 ...

🌟 Mersenne Primes

Primes of the form 2ⁿ - 1 less than 677

M₂ = 3 (n=2)

M₃ = 7 (n=3)

M₅ = 31 (n=5)

M₇ = 127 (n=7)

🎭 Palindromic Primes

Primes that read the same forwards and backwards

101

131

151

181

191

313

353

🔄 Circular Primes

Primes that remain prime when digits are rotated (below 677)

2 (single digit)

3 (single digit)

5 (single digit)

7 (single digit)

11 → 11, 11

13 → 13, 31

17 → 17, 71

31 → 31, 13

37 → 37, 73

71 → 71, 17

73 → 73, 37

79 → 79, 97

97 → 97, 79

113 → 113, 131, 311

131 → 131, 311, 113

197 → 197, 971, 719

199 → 199, 991, 919

311 → 311, 113, 131

337 → 337, 373, 733

373 → 373, 733, 337

🔢 Understanding Prime Numbers

Prime numbers are the building blocks of mathematics, encryption, and more. Explore what makes them fascinating.

What is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers cannot be formed by multiplying two smaller natural numbers. The first few are: 2, 3, 5, 7, 11, 13, 17, 19, 23...

🌍 Real-World Uses

🔐 RSA Encryption: Internet security
📱 Blockchain: Bitcoin mining
🎲 Random Numbers: Gaming algorithms
💻 Hash Tables: Database optimization
🛡️ Digital Signatures: Authentication
📡 Error Detection: Data transmission

🧮 Math Properties

2 is the only even prime
All primes >3 are 6k±1
Infinite quantity (Euclid's proof)
Fundamental Theorem of Arithmetic
Prime Number Theorem (density)
Goldbach's Conjecture

📚 Historical Facts

🏛️ Known since ancient Greeks
📜 Sieve of Eratosthenes (276 BC)
🧠 Fermat's Little Theorem (1640)
🎯 Euler's work on primes (1700s)
🏆 Riemann Hypothesis (1859)
💰 $1M Clay Prize problems

🏆 Modern Records

🥇 Largest: 2⁸²,⁵⁸⁹,⁹³³ - 1
📏 24,862,048 digits long
🖥️ Found by GIMPS project
⚡ Distributed computing
🎖️ $150,000 prize awarded
🔍 Search continues daily

🎓 Advanced Concepts

Prime Number Theorem

π(n) ≈ n/ln(n) - The density of primes decreases logarithmically

Riemann Hypothesis

ζ(s) = 0 only at s = -2,-4,-6... and Re(s) = 1/2

Bertrand's Postulate

For n > 1, there's always a prime between n and 2n

🔬 Testing Algorithms

Trial Division

Test divisibility up to √n - Simple but slow O(√n)

Miller-Rabin

Probabilistic test - Fast O(k log³ n) with k rounds

🔢 Prime Number Checker