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Reference & Checker

Prime Numbers

A whole number with exactly two factors — 1 and itself. Check any number instantly with step-by-step prime factorisation, play the interactive 1–100 Sieve game, or search the first 1,000 primes.

PRIMALITY CHECKER

Quick examples to test:

Step-by-step solution

Enter a number above to test primality and view step-by-step working.

What is a prime number?

A prime number is a whole number greater than 1 that has exactly two distinct positive factors: 1 and itself. It cannot be divided evenly by any other whole number.

7 ✓ factors: 1, 7
9 ✗ 3 × 3 = 9
1 ✗ only 1 factor

Key rule: 2 is the only even prime number — every other even number is divisible by 2.

Twin prime pairs

Pairs of prime numbers that differ by exactly 2 (p, p + 2):

Prime counting function π(n)

π(10)
4 primes
π(100)
25 primes
π(1,000)
168 primes
π(10,000)
1,229 primes
π(100,000)
9,592 primes
π(1,000,000)
78,498 primes

Prime Numbers Explained

Core concepts, classroom applications, and plain-English explanations for all audiences.

The atoms of all numbers

Just like all matter in chemistry is built from elements on the periodic table, every whole number greater than 1 is built uniquely by multiplying prime numbers together (the Fundamental Theorem of Arithmetic). For example, 12 is always 2 × 2 × 3.

Worked example: Is 91 prime?

"Many people assume 91 is prime because it is odd and doesn't end in 5. How do we test it?"

Find √91 ≈ 9.54 → only test primes up to 9: 2, 3, 5, 7

91 ÷ 2 = 45.5 (not divisible)

91 ÷ 3 = 30.33 (not divisible, sum 9+1=10)

91 ÷ 5 = 18.2 (doesn't end in 0 or 5)

91 ÷ 7 = 13 (exact whole number!)

Result: 91 = 7 × 13 (Composite, not prime)

Shortcut: you only ever need to test prime divisors up to the square root of the number.

Core rules to remember

  • → 1 is NOT prime: A prime must have exactly two distinct factors (1 and itself). 1 only has one factor.
  • → 2 is the ONLY even prime: Every other even number can be divided by 2, so it has at least 3 factors.

The Sieve of Eratosthenes

Invented around 200 BC, the Sieve finds all primes by writing out numbers 1 to 100, crossing out 1, circling 2 and crossing out all multiples of 2, circling 3 and crossing out all multiples of 3, and repeating. What's left uncrossed are the 25 primes under 100.

Interactive Game

Find the Primes (1 to 100)

Click every prime number in the grid. There are 25 primes between 1 and 100. Can you find them all?

25 primes left to find
0 found ✓ 0 wrong ✗
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Click a number — if it's prime it will light up ✓
Reference Table

First 1,000 Prime Numbers

Click any prime to test it in the checker above. The 1,000th prime is 7,919.

# +0+1+2+3+4+5+6+7+8+9
1–10 2 3 5 7 11 13 17 19 23 29
11–20 31 37 41 43 47 53 59 61 67 71
21–30 73 79 83 89 97 101 103 107 109 113
31–40 127 131 137 139 149 151 157 163 167 173
41–50 179 181 191 193 197 199 211 223 227 229
51–60 233 239 241 251 257 263 269 271 277 281
61–70 283 293 307 311 313 317 331 337 347 349
71–80 353 359 367 373 379 383 389 397 401 409
81–90 419 421 431 433 439 443 449 457 461 463
91–100 467 479 487 491 499 503 509 521 523 541
101–110 547 557 563 569 571 577 587 593 599 601
111–120 607 613 617 619 631 641 643 647 653 659
121–130 661 673 677 683 691 701 709 719 727 733
131–140 739 743 751 757 761 769 773 787 797 809
141–150 811 821 823 827 829 839 853 857 859 863
151–160 877 881 883 887 907 911 919 929 937 941
161–170 947 953 967 971 977 983 991 997 1009 1013
171–180 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069
181–190 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
191–200 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223
201–210 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291
211–220 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373
221–230 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451
231–240 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511
241–250 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583
251–260 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
261–270 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733
271–280 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811
281–290 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889
291–300 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987
301–310 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053
311–320 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129
321–330 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
331–340 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287
341–350 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357
351–360 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423
361–370 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531
371–380 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617
381–390 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687
391–400 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
401–410 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819
411–420 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903
421–430 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999
431–440 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079
441–450 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181
451–460 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257
461–470 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
471–480 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413
481–490 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511
491–500 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571
501–510 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643
511–520 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727
521–530 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821
531–540 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
541–550 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989
551–560 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057
561–570 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139
571–580 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231
581–590 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297
591–600 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409
601–610 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
611–620 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583
621–630 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657
631–640 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751
641–650 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831
651–660 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937
661–670 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003
671–680 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087
681–690 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179
691–700 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279
701–710 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387
711–720 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443
721–730 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521
731–740 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639
741–750 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693
751–760 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791
761–770 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857
771–780 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939
781–790 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053
791–800 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133
801–810 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221
811–820 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301
821–830 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367
831–840 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473
841–850 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571
851–860 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673
861–870 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761
871–880 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833
881–890 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917
891–900 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997
901–910 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103
911–920 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207
921–930 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297
931–940 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411
941–950 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499
951–960 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561
961–970 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643
971–980 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723
981–990 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829
991–1000 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919
Prime Curiosities

What makes primes so special?

∞

There are infinitely many primes

Euclid proved this around 300 BC. Assume a finite list of all primes p₁, p₂…pₙ. Multiply them all and add 1. The result is either prime or divisible by a prime not in the list, proving the primes can never end.

🏆

Largest known prime

2136,279,841 − 1 — a Mersenne prime with 41,024,320 digits. Found in October 2024 by GIMPS. Mersenne primes have the form 2p − 1 where p is itself prime.

🟡

Goldbach's conjecture (unproven)

"Every even integer greater than 2 is the sum of two primes." Stated in 1742, verified for numbers up to 4 × 10¹⁸, but still never proven.

10 = 3 + 7   28 = 5 + 23

🔐

Primes power internet security

RSA encryption relies on the asymmetry that multiplying two large primes is instant, but factoring their product back into primes is computationally infeasible with classical computers.

👯

Twin prime conjecture (unproven)

Are there infinitely many twin prime pairs (p, p+2)? Mathematicians strongly believe so, but a complete proof remains one of the holy grails of number theory.

🏗️

Fundamental theorem of arithmetic

Every integer greater than 1 is either prime or can be written as a unique product of primes. Primes are the fundamental building blocks (atoms) of arithmetic.

360 = 2³ × 3² × 5 (unique)