Displaying 1-10 of 11 results found.
Primes p such that the polynomial x^2+x+p generates only primes for x=1..11.
+10
8
17, 41, 1761702947, 8776320587, 10102729577, 11085833111, 177558051107, 273373448057, 473787509537, 557149355507, 715464238661, 1359854730821, 2131528031441, 2170341748697, 2236159108277, 2308235320997, 2751203698151, 3247566894821, 3288002848511, 3424305123047
Primes p such that the polynomial x^2+x+p generates only primes for x=1..10.
+10
7
17, 41, 180078317, 1278189947, 1761702947, 1829187287, 5862143447, 6369321857, 7226006861, 8776320587, 10102729577, 11085833111, 12412643261, 50626299797, 53039299211, 72355485857, 74621287901, 76233413141, 81948881447, 115826556611, 129077263697
PROG
(PARI) isokp(p) = {for (n=1, 10, if (! isprime(subst(x^2+x+p, x, n)), return (0)); ); 1; }
lista(nn) = {forprime (p=1, nn, if (isokp(p), print1(p, ", ")); ); } \\ Michel Marcus, Jan 05 2015
Primes of the form k^2 + k + 844427.
+10
5
844427, 844429, 844433, 844439, 844447, 844457, 844469, 844483, 844499, 844517, 844609, 844733, 844769, 844847, 845027, 845129, 845183, 845357, 845833, 845909, 845987, 846067, 846149, 846233, 846407, 846589, 846779, 846877, 846977, 847079, 847507, 847967, 848087
PROG
(Magma) [n^2+n+844427 : n in [0..60] | IsPrime(n^2+n+844427)]; // Bruno Berselli, Feb 23 2011 (PARI) for(n=0, 60, if(isprime(x=(n^2+n+844427)), print1(x, ", "))); \\ Arkadiusz Wesolowski, Mar 02 2011
Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.
+10
4
43, 47, 53, 71, 83, 113, 131, 173, 251, 281, 383, 461, 503, 593, 743, 73361, 73421, 3071069, 15949847, 76553693, 2204597, 1842719, 246407807, 986578883, 73975907, 4069235123, 1244414939, 25213427, 656856899, 30641069183, 8221946477, 41730358853, 10066886927, 285340609997, 6232338461
COMMENTS
This is a subsequence of A273756 which considers all odd numbers (2n+1) instead of only prime(n) as coefficients of the linear term. All terms are necessarily prime, since this is necessary and sufficient to get a prime for x = 0.
The respective minima (= number of consecutive primes for x = 0, 1, 2, ...) are given in A273597. It has been pointed out by Don Reble that the prime k-tuple conjecture predicts infinitely long sequences of primes of the given form, therefore we consider the "local" maxima, for q below some appropriate (large) limit: see sequences A273756 & A273770 for further details. - M. F. Hasler, Feb 17 2020
Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.
+10
4
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 73303, 73361, 73421, 73483, 3443897, 3071069, 3071137, 15949847, 76553693, 365462323, 365462399, 2204597, 9721, 1842719, 246407633, 246407719, 246407807, 246407897, 246407989
COMMENTS
All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, .... Sequence A273595 is the subsequence of terms for which 2n+1 is prime. For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)
PROG
(PARI) A273756(n, p=2*n+1, L=10^(5+n\10), m=0, Q)={forprime(q=1, L, for(x=1, oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q, m=x]; break)); Q}
EXTENSIONS
a(27) corrected and more terms from Don Reble, Feb 15 2018
Max { min { x >= 0 | p + (2*n+1)*x + x^2 is composite }, p < 10^(5+n/10) }.
+10
4
40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 12, 11, 10, 11, 12, 11, 12, 12, 13, 12, 12, 13, 16, 17, 16, 15, 14, 13, 13, 12, 11, 12, 13, 13, 14, 13, 13, 13, 12, 13, 14, 13, 14, 15, 14, 14, 13, 14, 14, 13
COMMENTS
The values for p are given in A273756 which is the main entry, see there for further information and (cross)references. From the initial values, the sequence seems strictly decreasing, with a(n) = 40-n, however, this property does not persist beyond a(27) = 13.
The upper limit on p ensures that we have a well-defined sequence: The prime k-tuple conjecture predicts existence of arbitrarily long sequences of primes of the given form, and thus unbounded minimal value of x. However, the corresponding prime tuples are expected to appear for much larger values of p. The given limit should be understood as "below the first/next such prime tuple", and in general the values a(n) should not change if that limit would be increased by some orders of magnitude. There might be counterexamples, which would be interesting. The given limit was chosen for lack of a more natural expression, and is relatively small. It could be replaced by a more appropriate function of n if a proposal is available, which should not affect the values given so far. - M. F. Hasler, Jan 22 2018, edited Feb 17 2020
FORMULA
a(n) = 40 - n for 0 <= n <= 27.
PROG
(PARI) { A273770(n, p=2*n+1, L=10^(5+n/10), m)=forprime(q=1, L, for(x=1, oo, ispseudoprime(q+p*x+x^2) || (x>m && !m=x) || break)); m}
EXTENSIONS
Corrected and extended by Don Reble, Feb 15 2018
Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.
+10
3
7, 43, 79, 457, 877, 967, 1093, 2437, 2683, 3187, 5077, 5923, 7933, 8233, 11923, 12889, 15787, 17389, 19993, 31543, 41113, 41617, 42457, 71359, 77863, 80683, 91393, 101719, 102643, 105967, 107347, 120163, 129733, 137593, 151783, 170263, 175723, 197569, 210127
EXAMPLE
a(1) = 7:
0^4 + 0^3 + 0^2 + 0 + 7 = 7;
1^4 + 1^3 + 1^2 + 1 + 7 = 11;
2^4 + 2^3 + 2^2 + 2 + 7 = 37;
3^4 + 3^3 + 3^2 + 3 + 7 = 127;
4^4 + 4^3 + 4^2 + 4 + 7 = 347;
5^4 + 5^3 + 5^2 + 5 + 7 = 787;
all six are primes.
MAPLE
select(p -> andmap(isprime, [p, p+4, p+30, p+120, p+340, p+780]), [seq(6*i+1, i=1..10^5)]); # Robert Israel, Jan 11 2015
MATHEMATICA
Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[20000]], AllTrue[#+{4, 30, 120, 340, 780}, PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)
PROG
(PARI) forprime(p=1, 500000, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780), print1(p, ", ")))
Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...6.
+10
3
43, 457, 967, 1093, 5923, 8233, 11923, 15787, 41113, 80683, 151783, 210127, 213943, 294919, 392737, 430879, 495559, 524827, 537007, 572629, 584557, 711727, 730633, 731593, 1097293, 1123879, 1138363, 1149163, 1396207, 1601503, 1739557, 1824139, 2198407, 2223853
EXAMPLE
a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 = 43;
1^4 + 1^3 + 1^2 + 1 + 43 = 47;
2^4 + 2^3 + 2^2 + 2 + 43 = 73;
3^4 + 3^3 + 3^2 + 3 + 43 = 163;
4^4 + 4^3 + 4^2 + 4 + 43 = 383;
5^4 + 5^3 + 5^2 + 5 + 43 = 823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
all seven are primes.
MATHEMATICA
Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5, 6}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[200000]], AllTrue[#+{4, 30, 120, 340, 780, 1554}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 10 2017 *)
PROG
(PARI) forprime(p=1, 1e6, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)& isprime(p+1554), print1(p, ", ")))
Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...7.
+10
3
43, 457, 967, 11923, 15787, 41113, 213943, 294919, 392737, 430879, 524827, 572629, 730633, 1097293, 1149163, 2349313, 2738779, 3316147, 3666007, 5248153, 5396617, 5477089, 7960009, 9949627, 10048117, 11260237, 11613289, 15281023, 16153279, 17250367, 18733807
EXAMPLE
a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 = 43;
1^4 + 1^3 + 1^2 + 1 + 43 = 47;
2^4 + 2^3 + 2^2 + 2 + 43 = 73;
3^4 + 3^3 + 3^2 + 3 + 43 = 163;
4^4 + 4^3 + 4^2 + 4 + 43 = 383;
5^4 + 5^3 + 5^2 + 5 + 43 = 823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
7^4 + 7^3 + 7^2 + 7 + 43 = 2843;
all eight are primes.
MATHEMATICA
Select[f=k^4+k^3+k^2+k; k={0, 1, 2, 3, 4, 5, 6, 7}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[12*10^5]], AllTrue[#+{4, 30, 120, 340, 780, 1554, 2800}, PrimeQ]&] (* Harvey P. Dale, Apr 24 2022 *)
PROG
(PARI) forprime(p=1, 1e8, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)& isprime(p+1554)& isprime(p+2800), print1(p, ", ")))
Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields primes for k = 0..8, but not for k = 9.
+10
2
43, 967, 11923, 213943, 2349313, 3316147, 30637567, 33421159, 39693817, 49978447, 105963769, 143405887, 148248949, 153756073, 156871549, 172981279, 187310803, 196726693, 203625283, 211977523, 220825453, 268375879, 350968543, 357834283, 414486697, 427990369
COMMENTS
All the terms in this sequence are congruent to 1 (mod 3).
EXAMPLE
a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 = 43;
1^4 + 1^3 + 1^2 + 1 + 43 = 47;
2^4 + 2^3 + 2^2 + 2 + 43 = 73;
3^4 + 3^3 + 3^2 + 3 + 43 = 163;
4^4 + 4^3 + 4^2 + 4 + 43 = 383;
5^4 + 5^3 + 5^2 + 5 + 43 = 823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
7^4 + 7^3 + 7^2 + 7 + 43 = 2843;
8^4 + 8^3 + 8^2 + 8 + 43 = 4723;
all nine are primes, and
9^4 + 9^3 + 9^2 + 9 + 43 = 7423 = 13*571 is composite.
The next prime for p=43 appears for k=13, namely 30983.
MATHEMATICA
Select[Prime[Range[118*10^5]], AllTrue[#+{0, 4, 30, 120, 340, 780, 1554, 2800, 4680}, PrimeQ]&&CompositeQ[#+7380]&] (* Harvey P. Dale, Sep 10 2021 *)
PROG
(PARI) forprime(p=1, 1e10, if(isprime(p+4)&& isprime(p+30)&& isprime(p+120)&& isprime(p+340)&& isprime(p+780)&& isprime(p+1554)&& isprime(p+2800)&& isprime(p+4680) && !isprime(p+7380), print1(p, ", ")))
CROSSREFS
Cf. A027445, A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458, A247949, A247966, A248206.
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