Delicate prime

A delicate prime, digitally delicate prime, or weakly prime number is a prime number where, under a given radix but generally decimal, replacing any one of its digits with any other digit always results in a composite number.^[1]

Definition[]

A prime number is called a digitally delicate prime number when, under a given radix but generally decimal, replacing any one of its digits with any other digit always results in a composite number.^[1] A weakly prime base-b number with n digits must produce $(b-1)\times n$ composite numbers after every digit is individually changed to every other digit. There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.^[2]

History[]

In 1978, Murray S. Klamkin posed the question of whether these numbers existed. Paul Erdős proved that there exists an infinite number of "delicate primes" under any base.^[1]

In 2007, Jens Kruse Andersen found the 1000-digit weakly prime $(17\times 10^{1000}-17)/99+21686652$ .^[3] This is the largest known weakly prime number as of 2011.

In 2011, Terence Tao proved in a 2011 paper, that delicate prime exist in a positive proportion for all bases.^[4] Positive proportion here means as the primes get bigger, the distance between the delicate primes will be quite similar, thus not scarce among prime numbers.^[1]

Examples[]

The smallest weakly prime base-b number for bases 2 through 10 are:^[5]


Base	In base	Decimal
2	1111111₂	127
3	2₃	2
4	11311₄	373
5	313₅	83
6	334155₆	28151
7	436₇	223
8	14103₈	6211
9	3738₉	2789
10	294001₁₀	294001

In the decimal number system, 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.

References[]

^ Jump up to: ^a ^b ^c ^d Nadis, Steve (30 March 2021). "Mathematicians Find a New Class of Digitally Delicate Primes". Quanta Magazine. Retrieved 2021-04-01.
^ Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society. 91 (3): 405–413. arXiv:0802.3361. doi:10.1017/S1446788712000043. S2CID 16931059.
^ Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. Retrieved 18 February 2011.
^ Tao, Terence (2010-04-18). "A remark on primality testing and decimal expansions". arXiv:0802.3361 [math.NT].
^ Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. Retrieved 18 February 2011.

[:12-1] Jump up to: ^a ^b ^c ^d Nadis, Steve (30 March 2021). "Mathematicians Find a New Class of Digitally Delicate Primes". Quanta Magazine. Retrieved 2021-04-01.

[2] Terence Tao (2011). "A remark on primality testing and decimal expansions". Journal of the Australian Mathematical Society. 91 (3): 405–413. arXiv:0802.3361. doi:10.1017/S1446788712000043. S2CID 16931059.

[3] Carlos Rivera. "Puzzle 17 – Weakly Primes". The Prime Puzzles & Problems Connection. Retrieved 18 February 2011.

[4] Tao, Terence (2010-04-18). "A remark on primality testing and decimal expansions". arXiv:0802.3361 [math.NT].

[5] Les Reid. "Solution to Problem #12". Missouri State University's Problem Corner. Retrieved 18 February 2011.

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hide v t Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) 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/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 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 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers