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Primes of the form x^3 + 2y^3, with x,y >0.
+10
12
3, 17, 29, 43, 127, 179, 251, 277, 359, 397, 433, 557, 593, 811, 857, 1051, 1367, 1459, 1583, 1753, 1801, 2017, 2027, 2213, 2251, 2447, 2663, 2689, 2729, 2789, 3221, 3331, 3391, 3457, 3581, 4421, 4519, 4787, 4967, 5653, 6037, 6217, 7109, 7883, 8081
COMMENTS
Heath-Brown shows that this sequence is infinite.
EXAMPLE
a(1) = 1^3+2*1^3 =3, prime. a(2) = 1^3 + 2* 2^3 = 17. a(7) = 1^3+2*r^3 =251.
MAPLE
T:=array(0..5000000): ind:=1: for x from 1 to 1000 do: for y from 1 to 1000 do: z:=x^3 + 2*y^3: if type(z, prime)=true then T[ind] :=z: ind :=ind+1: else fi: od: od: mini:=T[1]: ii:=1: for p from 1 to ind-1 do: for n from 1 to ind-1 do: if T[n] < mini then mini:= T[n]: ii:=n: else fi: od: print(mini): T[ii]:= 999999999999999: ii:=1: mini:=T[1] : od:
MATHEMATICA
formQ[p_] := Reduce[0 < x < p^(1/3) && 0 < y < (p/2)^(1/3) && x^3 + 2 y^3 == p, {x, y}, Integers] =!= False; Select[ Prime[ Range[1100]], formQ] (* Jean-François Alcover, Sep 28 2011 *)
PROG
(PARI) list(lim)=my(v=List(), t); for(y=1, sqrtn(lim\2, 3), t=2*y^3; for(x=1, sqrtn(lim-t, 3), if(isprime(t+x^3), listput(v, t+x^3)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Sep 28 2011
Primes which are the sum of three 5th powers.
+10
6
3, 307, 487, 9043, 16871, 17293, 17863, 23057, 32359, 32801, 33857, 36739, 40787, 43669, 50599, 59051, 59113, 62417, 65537, 76099, 101267, 104149, 107777, 135893, 160073, 161053, 164419, 249107, 249857, 256609, 259733, 266663, 338909, 340649
COMMENTS
Primes in the sumset { A000584 + A000584 + A000584}. There must be an odd number of odd terms in the sum, either 3 odd terms (as with 3 = 1^5 + 1^5 + 1^5 and 487 = 1^5 + 3^5 + 3^5 and 59051 = 1^5 + 1^5 + 9^5) or two even terms and one odd term (as with 307 = 2^5 + 2^5 + 3^5 and 9043 = 3^5 + 4^5 + 6^5). The sum of two positive 5th powers ( A003347), other than 2 = 1^5 + 1^5, cannot be prime. - Jonathan Vos Post, Sep 24 2006
EXAMPLE
a(1) = 3 = 1^5 + 1^5 + 1^5.
a(2) = 307 = 2^5 + 2^5 + 3^5.
a(3) = 487 = 1^5 + 3^5 + 3^5.
a(4) = 9043 = 3^5 + 4^5 + 6^5.
a(5) = 16871 = 2^5 + 2^5 + 7^5.
a(6) = 17293 = 3^5 + 3^5 + 7^5.
MATHEMATICA
lim = 10^6; nn = Floor[(lim - 2)^(1/5)]; t = {}; Do[p = i^5 + j^5 + k^5; If[p <= lim && PrimeQ[p], AppendTo[t, p]], {i, nn}, {j, i}, {k, j}]; t = Union[t] (* Vladimir Joseph Stephan Orlovsky and T. D. Noe, Jul 15 2011 *) Select[Prime[Range[2, 30000]], Length[PowersRepresentations[#, 3, 5]]>0&] (* Harvey P. Dale, Nov 26 2014 *)
Primes which are the sum of three nonzero 6th powers.
+10
3
3, 857, 1459, 4889, 50753, 51481, 66377, 119107, 210961, 262937, 308801, 525017, 531569, 539633, 562691, 766739, 797681, 840241, 1000793, 1046657, 1078507, 1772291, 1864873, 2303003, 2834443, 2986777, 3032641, 3107729, 3365777, 4757609, 4804201, 5135609, 5987593, 7530329, 7534361, 7743529, 8061041
COMMENTS
Primes of form x^6 + y^6 + z^6 where x, y, z > 0.
EXAMPLE
3 = 1^6 + 1^6 + 1^6;
857 = 2^6 + 2^6 + 3^6;
1459 = 1^6 + 3^6 + 3^6, etc.
MAPLE
N:= 10^8: # to get all terms <= N
S:= [seq(i^6, i=1..floor(N^(1/6)))]:
S3:= {seq(seq(seq(S[i]+S[j]+S[k], k=1..j), j=1..i), i=1..nops(S))}:
sort(convert(select(t -> t <= N and isprime(t), S3), list)); # Robert Israel, Mar 09 2017
MATHEMATICA
nn = 15; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^6)], # <= nn^6 && PrimeQ[#] &]
PROG
(PARI) list(lim)=my(v=List(), a6, a6b6, t); lim\=1; for(a=1, sqrtnint(lim-2, 6), a6=a^6; for(b=1, min(sqrtnint(lim-a6-1, 6), a), a6b6=a6+b^6; forstep(c=if(a6b6%2, 2, 1), min(sqrtnint(lim-a6b6, 6), b), 2, if(isprime(t=a6b6+c^6), listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Mar 09 2017
Primes which are the sum of three positive 7th powers.
+10
3
3, 257, 82499, 823799, 1119863, 2099467, 4782971, 5063033, 5608699, 6880249, 7160057, 10018571, 10078253, 10094509, 10279937, 10389481, 10823671, 19503683, 20002187, 20388839, 24782969, 31584323, 35850379, 36189869, 37931147, 50614777, 57416131, 62765029, 64845797, 68355029, 71663617, 73028453
COMMENTS
Primes of form x^7 + y^7 + z^7 where x, y, z > 0.
EXAMPLE
3 = 1^7 + 1^7 + 1^7;
257 = 1^7 + 2^7 + 2^7;
82499 = 3^7 + 3^7 + 5^7, etc.
MAPLE
N:= 10^9: # to get all terms <= N
Res:= {}:
for x from 1 to floor(N^(1/7)) do
for y from 1 to min(x, floor((N-x^7)^(1/7))) do
for z from 1 to min(y, floor((N-x^7-y^7)^(1/7))) do
p:= x^7 + y^7 + z^7;
if isprime(p) then Res:= Res union {p} fi
od od od:
MATHEMATICA
nn = 14; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^7)], # <= nn^7 && PrimeQ[#] &]
PROG
(PARI) list(lim)=my(v=List(), x7, y7, t, p); for(x=1, sqrtnint(lim\3, 7), x7=x^7; for(y=x, sqrtnint((lim-x7)\2, 7), y7=y^7; t=x7+y7; forstep(z=y+(x+1)%2, sqrtnint((lim-t)\1, 7), 2, if(isprime(p=t+z^7), listput(v, p))))); Set(v) \\ Charles R Greathouse IV, Feb 27 2017
Primes which are the sum of three nonzero 8th powers.
+10
3
3, 6563, 72353, 137633, 787811, 1745153, 7444673, 44726593, 49202147, 61503553, 86093443, 91858243, 100006817, 100072097, 101686177, 107444417, 143046977, 200006561, 214756067, 257412163, 300452323, 430372577, 431661313, 435812033, 447149537, 452523713, 489805633, 530372321, 744340577
COMMENTS
Primes of form x^8 + y^8 + z^8 where x, y, z > 0.
EXAMPLE
3 = 1^8 + 1^8 + 1^8;
6563 = 1^8 + 1^8 + 3^8;
72353 = 2^8 + 3^8 + 4^8, etc.
MATHEMATICA
nn = 13; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^8)], # <= nn^8 && PrimeQ[#] &]
PROG
(PARI) list(lim)=my(v=List(), A, B, t); lim\=1; for(a=1, sqrtnint(lim-2, 8), A=a^8; for(b=1, min(sqrtnint(lim-A-1, 8), a), B=A+b^8; forstep(c=if(B%2, 2, 1), sqrtnint(lim-B, 8), 2, if(isprime(t=B+c^8), listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017
Primes of the form p^3 + q^3 + r^3, where p, q and r are primes.
+10
2
43, 179, 277, 359, 397, 593, 811, 1483, 2017, 2213, 2251, 2447, 2689, 4421, 4519, 4967, 5381, 6271, 7109, 7229, 9181, 9521, 10169, 11897, 12853, 13103, 13841, 14489, 16561, 17107, 20357, 24443, 24677, 25747, 26711, 27917, 30161, 30259, 31193, 31247, 32579, 36161
COMMENTS
a(n) is a subset of A007490(n) = {3, 17, 29, 43, 73, 127, 179, 197, 251, 277, ...}, i.e., primes of the form x^3 + y^3 + z^3.
EXAMPLE
a(1) = 43 because 43 = 2^3 + 2^3 + 3^3 is prime and 2^3 + 2^3 + 2^3 = 24 is composite.
MATHEMATICA
lst={}; Do[Do[Do[p=n^3+m^3+k^3; If[PrimeQ[p]&&PrimeQ[n]&&PrimeQ[m]&&PrimeQ[k], AppendTo[lst, p]], {n, 4!}], {m, 4!}], {k, 4!}]; Take[Union[lst], 16] (* Vladimir Joseph Stephan Orlovsky, May 23 2009 *) With[{nn=40}, Select[Total/@Tuples[Prime[Range[nn]]^3, 3], PrimeQ[#]&&#<= nn^3+ 16&]]//Union (* Harvey P. Dale, Sep 08 2021 *)
CROSSREFS
Cf. A007490 = Primes of form x^3 + y^3 + z^3.
Twin primes both of which are the sum of three positive cubes.
+10
1
2267, 2269, 3527, 3529, 10331, 10333, 14867, 14869, 17207, 17209, 18521, 18523, 18917, 18919, 20231, 20233, 20357, 20359, 25577, 25579, 27791, 27793, 28547, 28549, 31247, 31249, 35279, 35281, 36899, 36901, 40697, 40699, 44279, 44281, 48779, 48781, 51479, 51481
EXAMPLE
3527 and 3529 are terms since 3527=3^3+5^3+15^3 and 3529=1^3+11^3+13^3.
MATHEMATICA
cu[n_] := {}!=Quiet@ IntegerPartitions[n, {3}, Range[n^(1/3)]^3, 1]; Flatten@ Rest@ Reap@ Do[If[ PrimeQ[p+2] && cu[p] && cu[p+2], Sow[{p, p+2}]], {p, Prime@ Range@ 10000}] (* Giovanni Resta, Apr 28 2016 *)
PROG
(PARI) list(lim)=my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y), if(isprime(t=k+z^3), listput(v, t))))); v=Set(v); for(i=2, #v-1, if(v[i]!=v[i-1]+2 && v[i]!=v[i+1]-2, v[i]=0)); v=Set(v); v[3..#v] \\ Charles R Greathouse IV, Apr 29 2016
Prime powers p^k such that p^k = x^3 + y^3 + z^3 where x, y, z are positive integers and k > 1, is soluble.
+10
0
81, 729, 2187, 2809, 3481, 5041, 6859, 14641, 15625, 19683, 24389, 26569, 27889, 59049, 63001, 68921, 83521, 148877, 273529, 300763, 332929, 357911, 375769, 413449, 531441, 597529, 619369, 657721, 683929, 704969, 707281, 744769, 776161, 779689, 844561, 877969, 912673
COMMENTS
Obviously, this sequence is infinite.
The first terms of this sequence are 3^4, 3^6, 3^7, 53^2, 59^2, 71^2, 19^3, 11^4, 5^6, 3^9, 29^3, 163^2, 167^2, 3^10, ...
EXAMPLE
81 is a term because 81 = 3^4 = 3^3 + 3^3 + 3^3.
MATHEMATICA
Select[Range[10^6], And[! PrimeQ@ #, PrimePowerQ@ #, Length[PowersRepresentations[#, 3, 3] /. {0, __} -> Nothing] > 0] &] (* Michael De Vlieger, Apr 17 2016 *)
PROG
(PARI) list(lim) = my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y), if(isprimepower(k+z^3) && !isprime(k+z^3), listput(v, k+z^3))))); Set(v);
Primes of form x^3 - (x + 1)^3 + 3*z^3 or -x^3 + (x + 1)^3 - 3*z^3, with x,z >= 0.
+10
0
2, 5, 7, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 73, 79, 101, 103, 107, 113, 127, 131, 139, 149, 173, 179, 181, 191, 193, 199, 251, 263, 269, 271, 277, 307, 317, 331, 367, 373, 379, 383, 389, 397, 431, 439, 479, 503, 509, 521, 523, 547, 557, 563, 569, 571
COMMENTS
For z <= 10^6, no other prime have this form in the first 105 primes.
EXAMPLE
0^3 - 1^3 + 3*2^3 = 23, 23 is a term.
-3^3 + 4^3 - 3*0^3 = -4^3 + 5^3 - 3*2^3 = -52^3 + 53^3 - 3*14^3 = 37, 37 is a term.
MATHEMATICA
p1 = Select[Prime[Range[105]], IntegerQ[(# - 1)/3] &];
p2 = Select[Prime[Range[105]], IntegerQ[(# + 1)/3] &];
n1 = Length@p1; n2 = Length@p2;
r1 = (p1 - 1)/3; r2 = (p2 + 1)/3;
t = {};
Do[x = (z^3 + r1[[n]] + 1/4)^(1/2) - 1/2;
If[IntegerQ[x], AppendTo[t, -x^3 + (x + 1)^3 - 3z^3]], {n, 1,
n1}, {z, 0, 270}]
Do[x = (z^3 - r2[[n]] + 1/4)^(1/2) - 1/2;
If[IntegerQ[x], AppendTo[t, x^3 - (x + 1)^3 + 3z^3]], {n, 1,
n2}, {z, 0, 170}]
Union@t
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