A237641 -id:A237641 - cp's OEIS frontend

A238136 Primes p such that p^4-p^3+1 and p^4-p^3-1 are also primes.

1429, 5827, 7411, 9601, 12601, 18457, 20011, 20521, 24919, 25999, 28591, 29947, 33211, 33349, 36037, 38149, 41227, 42649, 43579, 45307, 46099, 49999, 52057, 52387, 54319, 59107, 59197, 59629, 67891, 70951, 73477, 74761, 75037, 81157, 92041, 93607, 114889

Offset: 1

K. D. Bajpai, Feb 18 2014

nonn

			1429 is in the sequence because 1429, (1429^4-1429^3+1) and (1429^4-1429^3-1) are all primes.

K. D. Bajpai, Table of n, a(n) for n = 1..2918

Cf. A000040, A237639, A237641, A237642.

KD := proc() local a, b,d; a:=ithprime(n); b:= a^4-a^3+1;d:=a^4-a^3-1; if isprime (b) and isprime(d) then RETURN (a); fi; end: seq(KD(), n=1..20000);

Select[Prime[Range[3000]],PrimeQ[#^4-#^3+1]&&PrimeQ[#^4-#^3-1]&]
c=0;a=2;Do[k=Prime[n];  If[PrimeQ[k^4-k^3+1] &&PrimeQ[k^4-k^3-1], c=c+1;  Print[c," ",k]],  {n,1,2000000}];
pQ[n_]:=Module[{c=n^4-n^3},AllTrue[c+{1,-1},PrimeQ]]; Select[Prime[ Range[ 11000]],pQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 19 2014 *)

s=[]; forprime(p=2, 120000, if(isprime(p^4-p^3+1) && isprime(p^4-p^3-1), s=concat(s, p))); s \\ Colin Barker, Feb 18 2014

A237995 Primes p such that p^4 - p^3 - 1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 53, 59, 101, 103, 151, 157, 167, 193, 197, 239, 353, 379, 397, 419, 433, 467, 479, 503, 599, 641, 659, 661, 743, 787, 881, 907, 911, 983, 1049, 1109, 1123, 1153, 1201, 1229, 1291, 1307, 1373, 1399, 1429, 1531, 1601, 1621, 1663, 1747, 1753

Offset: 1

K. D. Bajpai, Feb 16 2014

nonn

			5 is in the sequence because 5 is prime and 5^4 - 5^3 - 1 = 499 is also prime.
17 is in the sequence because 17 is prime and 17^4 - 17^3 - 1 = 78607 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..3700

Cf. A000040, A237639, A237641, A237642.

KD := proc() local a,b; a:= ithprime(n); b:= a^4-a^3-1;if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..400);

c = 0; a = 2; Do[k = Prime[n]; If[PrimeQ[k^4 - k^3 - 1], c = c + 1;  Print[c, " ", k]], {n, 100000}]; (* Bajpai *)
Select[Prime[Range[200]], PrimeQ[#^4 - #^3 - 1] &] (* Alonso del Arte, Feb 17 2014 *)

s=[]; forprime(p=2, 2000, if(isprime(p^4-p^3-1), s=concat(s, p))); s \\ Colin Barker, Feb 17 2014

A238083 Primes p such that p^4 - p^3 + 1 is also prime.

Original entry on oeis.org

67, 139, 337, 409, 577, 607, 631, 1297, 1321, 1429, 1459, 1549, 1627, 2377, 2557, 2767, 2851, 2917, 3001, 3187, 3319, 3499, 4027, 4099, 4621, 4861, 4969, 5059, 5431, 5449, 5581, 5827, 5857, 6007, 6037, 6379, 6481, 6781, 6997, 7411, 7927, 8089, 8191, 8311

Offset: 1

K. D. Bajpai, Feb 17 2014

nonn

			67 is in the sequence because 67 is prime and 67^4 - 67^3 + 1 = 19850359 is also prime.
337 is in the sequence because 337 is prime and 337^4 - 337^3 + 1 = 12859645009 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..3409

Cf. A000040, A237639, A237641, A237642.

KD := proc() local a,b;  a:= ithprime(n);  b:= a^4 - a^3 + 1;  if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..2000);

c=0; a=2; Do[k=Prime[n];  If[PrimeQ[k^4-k^3+1], c=c+1;  Print[c," ",k]],    {n,1,100000}];
Select[Prime[Range[1100]],PrimeQ[#^4-#^3+1]&] (* Harvey P. Dale, Jun 11 2025 *)

isok(p) = isprime(p) && isprime(p^4 - p^3 + 1); \\ Michel Marcus, Feb 27 2014

More terms from Michel Marcus, Feb 27 2014

cp's OEIS Frontend

A238136 Primes p such that p^4-p^3+1 and p^4-p^3-1 are also primes.

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A237995 Primes p such that p^4 - p^3 - 1 is also prime.

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A238083 Primes p such that p^4 - p^3 + 1 is also prime.

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