Displaying 1-10 of 14 results found.
2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499
COMMENTS
Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021). The name refers to the American mathematician William Harold Mills (1921-2007). - Amiram Eldar, Jun 23 2021
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
LINKS
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, Amer. Math. Monthly, Vol. 126, No. 1 (2019), pp. 72-73; ResearchGate link, arXiv preprint, arXiv:2010.15882 [math.NT], 2020.
EXAMPLE
a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - Jonathan Vos Post, May 05 2006
MAPLE
floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. ( A051021).
MATHEMATICA
p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
PROG
(PARI) apply( { A051254(n, p=2)=while(n--, p=nextprime(p^3)); p}, [1..6]) \\ M. F. Hasler, Sep 11 2024
CROSSREFS
Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913. Cf. A224845 (integer lengths of Mills primes). Cf. A108739 (sequence of offsets b_n associated with Mills primes). Cf. A051021 (decimal expansion of Mills constant).
Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.
+10
6
3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568, 8436308
COMMENTS
This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues. Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - Serge Batalov, Aug 06 2013
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
FORMULA
b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;
EXAMPLE
The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - M. F. Hasler, Jul 22 2013
MATHEMATICA
B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)
PROG
(PARI) p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020
EXTENSIONS
a(9)-a(11) from Caldwell and Cheng, Aug 29 2005
Numbers given by a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677... is the constant defined at A338837
+10
6
2, 3, 11, 71, 701, 9467, 168599, 3860009, 111498091, 4002608003, 176359202639, 9437436701437, 607818993573569, 46744099128452807, 4262700354254812091, 458091929703695291747, 57691186909930154615407, 8471601990692484416847631, 1443868262009075144775972529
COMMENTS
Assuming Cramer's conjecture on largest prime gaps, it can be proved that there exists at least one constant 'c' such that all a(n) are primes for n as large as required. The constant giving the smallest growth rate is c=2.2679962677067242473285532807253717745270422544...
This exponential sequence of prime numbers grows very slowly compared to Mills' sequence for which each new term has 3 times more digits than the previous one. More than 60 terms (all prime numbers) can be easily calculated for the sequence described here which is quite remarkable for an exponential sequence.
Algorithm to compute the smallest constant 'c' and the associated prime number sequence a(n).
0. n=0, a(0)=2, c=2, d=1.5
1. n=n+1
2. b=1+floor(c^(n^d))
3. p=smpr(b) smallest prime >= b
4. If p=b then a(n)=p, go to 1.
5. c=(p-1)^(1/n^d)
6. a(n)=p
7. k=1
8. b=1+floor(c^(k^d))
9. If b<>a(k) then p=smpr(b), n=k, go to 5.
10. If k<n-1 then k=k+1, go to 8.
11. go to 1.
I propose the following generalization: find the function f(n) with f(0)=0 and f(x)>x for x>=2 such that there exists a suitable positive constant c(f) giving the increasing prime sequence a(n)=1+floor(c^f(n)) with the smallest possible growth rate. Since a(0)=2, c(f)>=2.
FORMULA
a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677...
PROG
(PARI)
c(n=40, prec=100)={
my(curprec=default(realprecision));
default(realprecision, max(prec, curprec));
my(a=List([2]), d=1.5, c=2.0, b, p, ok, smpr(b)=my(p=b); while(!isprime(p), p=nextprime(p+1)); return(p); );
for(j=1, n-1,
b=1+floor(c^(j^d));
until(ok,
ok=1;
p=smpr(b);
listput(a, p, j+1);
if(p!=b,
c=(p-1)^(j^(-d));
for(k=1, j-2,
b=1+floor(c^(k^d));
if(b!=a[k+1],
ok=0;
j=k;
break;
);
);
);
);
);
default(realprecision, curprec);
return(a);
Generalized Mills numbers: a(n) = floor(c^(b^n)) where c=4.4, b=1.179.
+10
4
5, 7, 11, 17, 29, 53, 109, 252, 679, 2184, 8650, 43828, 296913, 2832896, 40474353, 930818760, 37522518949, 2931502379404, 499688559138590, 213681340556825047, 270268120176240462240, 1227682581046707804164120
COMMENTS
First seven terms are primes.
MAPLE
Digits := 100; A060449 := n->4.4^(1.179^n);
PROG
(PARI) { default(realprecision, 2000); for (n=1, 44, write("b060449.txt", n, " ", floor(4.4^(1.179^n))); ) } \\ Harry J. Smith, Jul 05 2009
a(n) = floor(A^(C^n)), where A = 2.084551112207285611..., C = 1.221.
+10
4
2, 2, 3, 5, 7, 11, 19, 37, 83, 223, 739, 3181, 18911, 166679, 2376391, 60953117, 3202432763, 403823050201
COMMENTS
Results from the application of Caldwell's Generalized Mills's Theorem. This value of A produces 18 primes. For 20 primes A must be adjusted to 2.084551112207285611.
The extension of the sequence is guaranteed by the Cramer conjecture. That is: If the needed change in Y(n) for obtaining the next prime (superior or inferior) is as maximum = (log Y(n))^2/2, then the effect on Y(n-1) is less than K*C^(2n-1)*Y(n-1)/Y(n). K = (1/2)*(log A)^2 = 0.269784 This value diminishes with n. Example: For n = 23, a change in Y(23) by 2630 only changes Y(22) by 0.0043. _Jens Kruse Anderson_ with A = 2.084551112197624209091521123 calculated Y(n) = floor(A^(C^n)) from n = 1 to n = 3, obtaining 22 different primes. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009
REFERENCES
Jens Kruse Andersen. Personal communication (Feb 2009). [Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009]
O. Ore, Theory of Numbers and Its History. McGraw Hill, 1948.
FORMULA
a(n) = floor(A^(C^n)); A = 2.084551112... ; C = 1.221. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009
EXAMPLE
a(10) = 223 because 2.0845511122073^(1.221^10)= 223.58376...
With the value of A received from Jens K. Andersen we have: For n = 23, a(23) = 313 990 383 602 932 052 632 553 770 22009. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009
AUTHOR
Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Apr 20 2001
Decimal expansion of the largest C_0 = 1.2209864... such that for C < C_0 and A < 2 the sequence a(n) = floor[A^(C^n)] can't contain only prime terms.
+10
4
1, 2, 2, 0, 9, 8, 6, 4, 0, 7, 1, 3, 9, 5, 5, 0, 2, 4, 4, 2, 7, 3, 7, 0, 1, 4, 5, 1, 8, 8, 3, 5, 5, 8, 1, 4, 1, 6, 4, 6, 2, 4, 7, 5, 4, 0, 6, 0, 2, 9, 3, 8, 4, 4, 4, 7, 9, 1, 9, 7, 2, 9, 2, 5, 3, 7, 5, 1, 0, 3, 8, 7, 9, 7, 4, 6, 0, 0, 9, 1, 9, 1, 0, 3, 4, 2
COMMENTS
It is not proved that for C > C_0 the mentioned infinite sequence of primes actually exists. However, heuristics show that A243358 could be infinite (the decimal expansion of corresponding A value is A243370).
FORMULA
C_0 can be estimated as (logP/log84)^(1/k), where P is k+10th term of A243358.
Decimal expansion of Bertrand's constant.
+10
2
1, 2, 5, 1, 6, 4, 7, 5, 9, 7, 7, 9, 0, 4, 6, 3, 0, 1, 7, 5, 9, 4, 4, 3, 2, 0, 5, 3, 6, 2, 3, 3, 4, 6, 9, 6, 9
COMMENTS
From Bertrand's postulate (i.e., there is always a prime p in the range n < p < 2n) one can show there is a constant b such that floor(2^b), floor(2^2^b), ..., floor(2^2^2...^b), ... are all primes.
This result is due to Wright (1951), so Bertrand's constant might be better called Wright's constant, by analogy with Mills's constant A051021. - Jonathan Sondow, Aug 02 2013
REFERENCES
S. Finch, Mathematical Constants, Cambridge Univ. Press, 2003; see section 2.13 Mills's constant.
FORMULA
1.251647597790463017594432053623346969...
EXAMPLE
2^(2^(2^1.251647597790463017594432053623)) is approximately 37.0000000000944728917062132870071 and A051501(3)=37.
EXTENSIONS
More digits (from the Prime Curios page) added by Frank Ellermann, Sep 19 2011
Decimal expansion of x, where x is the smallest number for which floor[x^(2^y)] is prime for every y>0.
+10
2
1, 5, 2, 4, 6, 9, 9, 9, 6, 0, 5, 3, 8, 0, 9, 4, 3, 5, 9, 9, 2, 3, 3, 6, 3, 5, 7, 5, 6, 8, 8, 4, 2, 1, 1, 6, 2, 2, 2, 0, 2, 2, 3, 6, 2, 3, 1, 9, 9, 7, 7, 1, 2, 1, 9, 8, 4, 5, 7, 2, 2, 2, 6, 5, 6, 8, 4, 1, 6, 3, 0, 4, 8, 2, 0, 2, 3, 5, 7, 2, 3, 2, 3, 7, 0, 4, 8, 2, 3, 8, 3, 0, 4, 8, 6, 0, 9, 5, 7, 9, 5, 9, 9, 7, 6
EXAMPLE
1.524699960538...
Repeated squaring gives the primes 2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, ...
CROSSREFS
Cf. A051021, decimal expansion of Mills' constant where floor[x^(3^y)] is prime for every y>0.
Floor(A^(C^n)), where A = 32.76 and C = 1.33.
+10
2
103, 479, 3673, 55147, 2024063, 243937297, 142915724779, 685893080269745, 53978528420922581864, 175329092084368391071206608, 80227969100540338877503013472650510, 26469961649988241699181245714190498215773679043
COMMENTS
First seven terms are primes.
FORMULA
a(n) = floor(32.76^(1.33^n)).
EXAMPLE
a(2) = 479 because 32.76^(1.33^2) = 479.1724192479....
PROG
(PARI) default(realprecision, 100); for(n=1, 12, print1(floor(32.76^(1.33^n)), ", ")); \\ Arkadiusz Wesolowski, Jul 18 2011
Decimal expansion of what appears to be the smallest possible C for which the nearest integer to C^2^n is always prime and starts with 2.
+10
2
1, 2, 7, 2, 0, 1, 9, 6, 3, 3, 1, 9, 2, 1, 9, 3, 4, 9, 5, 8, 6, 9, 7, 3, 5, 3, 2, 0, 9, 1, 1, 9, 2, 8, 8, 3, 7, 6, 3, 7, 5, 6, 3, 0, 8, 2, 6, 9, 9, 6, 4, 7, 6, 4, 8, 1, 3, 2, 2, 5, 8, 0, 4, 1, 5, 4, 8, 7, 5, 3, 2, 8, 1, 4, 2, 6, 4, 3, 3, 7, 5, 6, 4, 0, 7, 3, 8, 4, 8, 8, 1, 5, 0, 4, 5, 1, 8, 7, 5, 4, 0, 7, 4, 0, 2, 8
COMMENTS
The square of this constant, C^2 = 1.6180339472264..., is very close to the Golden Ratio Phi ( A001622). This constant is about 3% less than Mills's constant, 1.306377883863..., ( A051021). Since there is always a prime between an integer and its square, this constant should satisfy the same criteria as does Mills's constant ( A051021). This constant, C, produces A059785.
EXAMPLE
=1.2720196331921934958697353209119288376375630826996476481322580415...
MATHEMATICA
RealDigits[ Nest[ NextPrime[#^2, -1] &, 2, 8]^(2^-9), 10, 111][[1]]
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