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Numbers n such that n-LargestCube is prime, (LargestCube <= n).
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3, 4, 6, 10, 11, 13, 15, 19, 21, 25, 29, 30, 32, 34, 38, 40, 44, 46, 50, 56, 58, 66, 67, 69, 71, 75, 77, 81, 83, 87, 93, 95, 101, 105, 107, 111, 117, 123, 127, 128, 130, 132, 136, 138, 142, 144, 148, 154, 156, 162, 166, 168, 172, 178, 184, 186, 192, 196, 198, 204
EXAMPLE
3-1^3=2, 4-1^3=3, ..., 10-2^3=2, 11-2^3=3, ..., 29-3^3=2, ....
MATHEMATICA
lst={}; Do[p=n-Floor[n^(1/3)]^3; If[PrimeQ[p], AppendTo[lst, n]], {n, 6!}]; lst
Select[Range[300], PrimeQ[#-Floor[Surd[#, 3]]^3]&] (* Harvey P. Dale, May 31 2017 *)
Prime numbers p such that p-LargestCube is prime, (LargestCube <= p).
+10
3
3, 11, 13, 19, 29, 67, 71, 83, 101, 107, 127, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 313, 317, 523, 541, 571, 601, 613, 619, 643, 661, 691, 709, 1013, 1019, 1031, 1061, 1097, 1103, 1109, 1151, 1163, 1181, 1193, 1223, 1229, 1277, 1283, 1307, 1733
COMMENTS
3-1^3=2, 11-2^3=3, 13-2^3=5, 29-3^3=2,..
MATHEMATICA
lst={}; Do[p=n-Floor[n^(1/3)]^3; If[PrimeQ[p]&&PrimeQ[n], AppendTo[lst, n]], {n, 7!}]; lst
Select[Prime[Range[300]], PrimeQ[#-Floor[Surd[#, 3]]^3]&] (* Harvey P. Dale, May 19 2019 *)
Primes p such that p+(floor(Sqrt(p)))^2 is prime.
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1
2, 7, 37, 43, 47, 67, 73, 149, 163, 167, 223, 337, 349, 353, 359, 409, 421, 439, 487, 499, 577, 587, 617, 691, 787, 823, 829, 911, 947, 1039, 1063, 1087, 1201, 1297, 1321, 1361, 1367, 1453, 1459, 1483, 1609, 1621, 1657, 1777, 1783, 1987, 1993, 2011, 2137, 2143
COMMENTS
2+1=3 prime, 7+4=11 prime, 37+36=73 prime,...
MATHEMATICA
f[n_]:=n+(Floor[Sqrt[n]])^2; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n, 7!}]; lst
Select[Prime[Range[400]], PrimeQ[#+Floor[Sqrt[#]]^2]&] (* Harvey P. Dale, Apr 24 2013 *)
Primes p such that p-+(floor(Sqrt(p)))^2 are primes.
+10
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7, 43, 47, 67, 149, 163, 167, 337, 353, 487, 587, 617, 787, 911, 947, 1367, 1777, 1783, 2333, 2347, 2503, 2927, 2953, 2963, 3023, 3607, 3613, 3637, 3643, 3697, 3709, 3847, 4363, 4397, 4423, 4463, 4483, 4903, 5273, 6113, 6143, 6197, 7103, 7187, 7193, 8117
MATHEMATICA
f1[n_]:=n-(Floor[Sqrt[n]])^2; f2[n_]:=n+(Floor[Sqrt[n]])^2; lst={}; Do[p=Prime[n]; If[PrimeQ[f1[p]]&&PrimeQ[f2[p]], AppendTo[lst, p]], {n, 8!}]; lst
fQ[n_]:=Module[{c=Floor[Sqrt[n]]^2}, AllTrue[n+{c, -c}, PrimeQ]]; Select[ Prime[ Range[1200]], fQ] (* Harvey P. Dale, Dec 15 2021 *)
Prime numbers p such that p-q^3 is a prime number, (q is a prime number, q^3=LargestCube, LargestCube <= p).
+10
1
11, 13, 19, 29, 127, 24391, 357913, 571789
COMMENTS
3-1^3=2, 11-2^3=3, 13-2^3=5, 19-2^3=11, 29-3^3=2, 127-5^3=2,..
MATHEMATICA
lst={}; Do[q=Floor[n^(1/3)]; p=n-q^3; If[PrimeQ[p]&&PrimeQ[n]&&PrimeQ[q], AppendTo[lst, n]], {n, 2*9!}]; lst
Prime numbers p such that p-LargestSquare is prime and p-LargestCube is also prime, (LargestSquare <= p, LargestCube <= p).
+10
1
3, 11, 19, 67, 71, 83, 107, 227, 263, 269, 613, 619, 1031, 1061, 1163, 1193, 1223, 1307, 1787, 1801, 1811, 1831, 1979, 1997, 2129, 4099, 4127, 4133, 4139, 4157, 4373, 4409, 4463, 4637, 4643, 4703, 5843, 5849, 5879, 5903, 6089, 6101, 6113, 6143, 6163, 6211
COMMENTS
11-3^2=2;11-2^3=3, 19-4^2=3,19-2^3=11,..
MATHEMATICA
lst={}; Do[p2=n-Floor[Sqrt[n]]^2; p3=n-Floor[n^(1/3)]^3; If[PrimeQ[p2]&&PrimeQ[p3]&&PrimeQ[n], AppendTo[lst, n]], {n, 8!}]; lst
plsplcQ[p_]:=AllTrue[{p-Floor[Sqrt[p]]^2, p-Floor[Surd[p, 3]]^3}, PrimeQ]; Select[ Prime[ Range[1000]], plsplcQ] (* Harvey P. Dale, Jul 03 2022 *)
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