Weakly prime number

Not to be confused with weak prime.

In number theory, a prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit.^[1] Decimal digits are usually assumed.

The first weakly prime numbers are:

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, ... (sequence A050249 in the OEIS)

For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b number with n digits must produce (b−1) × n composite numbers when a digit is changed.

In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17×10¹⁰⁰⁰−17)/99 + 21686652.^[2] This is the largest known weakly prime number as of 2011.

There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.^[3]

The smallest base b weakly primes for b = 1 to 16 are: (sequence A186995 in the OEIS) ^[4]

11₁ = 2

1111111₂ = 127

2₃ = 2

11311₄ = 373

313₅ = 83

334155₆ = 28151

436₇ = 223

14103₈ = 6211

3738₉ = 2789

294001₁₀ = 294001

2573₁₁ = 3347

6B8AB77₁₂ = 20837899

2216₁₃ = 4751

C371CD₁₄ = 6588721

9880C₁₅ = 484439

D2A45₁₆ = 862789

References

↑ Weisstein, Eric W. "Weakly Prime". MathWorld.
↑ Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. Retrieved 18 February 2011.
↑ Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society. 91 (3). arXiv:0802.3361. doi:10.1017/S1446788712000043.
↑ Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. Retrieved 18 February 2011.

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	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

List of prime numbers

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