RAMANUJAN PRIME - Encyclopedia Information

In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.

Origins and definition

In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev.^[1] At the end of the two-page published paper, Ramanujan derived a generalized result, and that is:

\pi (x)-\pi \left({\frac {x}{2}}\right)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}

OEIS: A104272

where $\pi (x)$ is the prime-counting function, equal to the number of primes less than or equal to x.

The converse of this result is the definition of Ramanujan primes:

The nth Ramanujan prime is the least integer R_n for which

\pi (x)-\pi (x/2)\geq n,

for all x ≥ R_n.^[2] In other words: Ramanujan primes are the least integers R_n for which there are at least n primes between x and x/2 for all x ≥ R_n.

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.

Note that the integer R_n is necessarily a prime number: $\pi (x)-\pi (x/2)$ and, hence, $\pi (x)$ must increase by obtaining another prime at x = R_n. Since $\pi (x)-\pi (x/2)$ can increase by at most 1,

\pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.

Bounds and an asymptotic formula

For all $n\geq 1$ , the bounds

2n\ln 2n<R_{n}<4n\ln 4n

hold. If $n>1$ , then also

p_{2n}<R_{n}<p_{3n}

where p_n is the nth prime number.

As n tends to infinity, R_n is asymptotic to the 2nth prime, i.e.,

R_n ~ p_2n (n → ∞).

All these results were proved by Sondow (2009),^[3] except for the upper bound R_n < p_3n which was conjectured by him and proved by Laishram (2010).^[4] The bound was improved by Sondow, Nicholson, and Noe (2011)^[5] to

R_{n}\leq {\frac {41}{47}}\ p_{3n}

which is the optimal form of R_n ≤ c·p_3n since it is an equality for n = 5.

References

^ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
^ Jonathan Sondow. "Ramanujan Prime". MathWorld.
^ Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv: 0907.5232, doi: 10.4169/193009709x458609
^ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, CiteSeerX 10.1.1.639.4934, doi: 10.1142/s1793042110003848.
^ Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps" (PDF), Journal of Integer Sequences, 14: 11.6.2, arXiv: 1105.2249, Bibcode: 2011arXiv1105.2249S

v t e 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
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 Unique
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 ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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 list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number 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

In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.

Origins and definition

\pi (x)-\pi \left({\frac {x}{2}}\right)\geq 1,2,3,4,5,\ldots {\text{ for all }}x\geq 2,11,17,29,41,\ldots {\text{ respectively}}

OEIS: A104272

where $\pi (x)$ is the prime-counting function, equal to the number of primes less than or equal to x.

The converse of this result is the definition of Ramanujan primes:

The nth Ramanujan prime is the least integer R_n for which

\pi (x)-\pi (x/2)\geq n,

for all x ≥ R_n.^[2] In other words: Ramanujan primes are the least integers R_n for which there are at least n primes between x and x/2 for all x ≥ R_n.

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.

\pi (R_{n})-\pi \left({\frac {R_{n}}{2}}\right)=n.

Bounds and an asymptotic formula

For all $n\geq 1$ , the bounds

2n\ln 2n<R_{n}<4n\ln 4n

hold. If $n>1$ , then also

p_{2n}<R_{n}<p_{3n}

where p_n is the nth prime number.

As n tends to infinity, R_n is asymptotic to the 2nth prime, i.e.,

R_n ~ p_2n (n → ∞).

R_{n}\leq {\frac {41}{47}}\ p_{3n}

which is the optimal form of R_n ≤ c·p_3n since it is an equality for n = 5.

References

^ Ramanujan, S. (1919), "A proof of Bertrand's postulate", Journal of the Indian Mathematical Society, 11: 181–182
^ Jonathan Sondow. "Ramanujan Prime". MathWorld.
^ Sondow, J. (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv: 0907.5232, doi: 10.4169/193009709x458609
^ Laishram, S. (2010), "On a conjecture on Ramanujan primes" (PDF), International Journal of Number Theory, 6 (8): 1869–1873, CiteSeerX 10.1.1.639.4934, doi: 10.1142/s1793042110003848.
^ Sondow, J.; Nicholson, J.; Noe, T.D. (2011), "Ramanujan primes: bounds, runs, twins, and gaps" (PDF), Journal of Integer Sequences, 14: 11.6.2, arXiv: 1105.2249, Bibcode: 2011arXiv1105.2249S

v t e 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
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 Unique
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 ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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 list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number 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

Origins and definition

Bounds and an asymptotic formula

References

Origins and definition

Bounds and an asymptotic formula

References

Videos

Websites

Encyclopedia

Facebook