A358743 -id:A358743 - cp's OEIS frontend

A358742 First of three consecutive primes p,q,r such that p^3 + q^3 - r^3 is prime.

13, 29, 89, 97, 127, 137, 151, 163, 199, 223, 241, 277, 313, 349, 367, 389, 419, 431, 457, 463, 521, 577, 613, 691, 823, 827, 829, 859, 877, 883, 911, 953, 971, 1049, 1087, 1097, 1129, 1151, 1163, 1217, 1409, 1489, 1499, 1579, 1699, 1723, 1867, 1879, 1993, 2089, 2111, 2141, 2293, 2339, 2399, 2411

Offset: 1

J. M. Bergot and Robert Israel, Nov 29 2022

nonn

Note: for x >= 3275, there is a prime between x and x(1 + 1/(2 log^2 x)) < 1.01x (Dusart 1998). Together with finite checking this shows that for p > 19, p^3 + q^3 - r^3 > 0. - Charles R Greathouse IV, Nov 29 2022

			a(3) = 89 is a term because 89, 97, 101 are consecutive primes and 89^3 + 97^3 - 101^3 = 587341 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A255581, A358743, A358744.

R:= NULL: count:= 0: q:= 2: r:= 3:
while count < 100 do
  p:= q; q:= r; r:=nextprime(r);
  if isprime(p^3+q^3-r^3) then count:= count+1; R:= R,p; fi;
od:
R;

Select[Partition[Prime[Range[360]], 3, 1], (s = #[[1]]^3 + #[[2]]^3 - #[[3]]^3) > 0 && PrimeQ[s] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)

a358742(upto) = {my(p1=2, p2=3); forprime(p3=5, upto, if (isprime (p1^3+p2^3-p3^3), print1(p1,", ")); p1=p2; p2=p3)};
a358742(2500) \\ Hugo Pfoertner, Nov 29 2022

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if isprime(p**3 + q**3 - r**3): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 56))) # Michael S. Branicky, Nov 29 2022

A358744 First of three consecutive primes p, q, r such that p + q - r, p^2 + q^2 - r^2 and p^3 + q^3 - r^3 are all prime.

Original entry on oeis.org

13, 29, 137, 521, 577, 691, 823, 1879, 3469, 4799, 8783, 21569, 25453, 26263, 26591, 27529, 27919, 34607, 39509, 45631, 48869, 53653, 56099, 56633, 57641, 63313, 63809, 67733, 68819, 74381, 76031, 76421, 94781, 97187, 98873, 101279, 105683, 110291, 118967, 119569, 119849, 120577, 123737, 128951

Offset: 1

J. M. Bergot and Robert Israel, Nov 29 2022

nonn

			a(3) = 137 is a term because 137, 139, 149 are consecutive primes and
137^1 + 139^1 - 149^1 = 127,
137^2 + 139^2 - 149^2 = 15889,
and 137^3 + 139^3 - 149^3 = 1949023 are all prime.

Robert Israel, Table of n, a(n) for n = 1..2500

Intersection of A358743, A255581 and A358742.

R:= NULL: count:= 0: q:= 2: r:= 3:
while count < 100 do
  p:= q; q:= r; r:=nextprime(r);
  if isprime(p+q-r) and isprime(p^2+q^2-r^2) and isprime(p^3+q^3-r^3) then count:= count+1; R:= R,p fi;
od:
R;

Select[Partition[Prime[Range[13000]], 3, 1], AllTrue[Table[#[[1]]^k + #[[2]]^k - #[[3]]^k, {k, 1, 3}], PrimeQ] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if all(isprime(t) for t in [p+q-r, p**2+q**2-r**2, p**3+q**3-r**3]):
            yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 44))) # Michael S. Branicky, Nov 29 2022

A358745 a(n) is the least prime p that is the first of three consecutive primes p, q, r such that p^i + q^i - r^i is prime for i from 1 to n but not n+1.

Original entry on oeis.org

2, 7, 41, 13, 4799, 45631, 332576273, 157108359787, 4001045161

Offset: 0

J. M. Bergot and Robert Israel, Nov 29 2022

For any prime x, if p == r (mod x) and q <> x, or q == r (mod x) and p <> x, p^i + q^i - r^i is not divisible by x. Thus there is no modular obstruction to the sequence being infinite.

If a(9) exists, then it exceeds 8*10^12. - Lucas A. Brown, Mar 08 2024

			a(3) = 13 because 13, 17, 19 are consecutive primes with 13 + 17 - 19 = 11, 13^2 + 17^2 - 19^2 = 97 and 13^3 + 17^3 - 19^3 = 251 are prime but 13^4 + 17^4 - 19^4 = -18239 is not, and no prime less than 13 works.

Cf. A255581, A358742, A358743, A358744.

V:= Array(0..5):
q:= 2: r:= 3: count:= 0:
while count < 6 do
  p:= q; q:= r; r:= nextprime(r);
  for i from 1 while isprime(p^i+q^i-r^i) do od:
  if V[i-1] = 0 then V[i-1]:= p; count:= count+1 fi;
od:
convert(V,list);

a(n) = my(p=2, q=nextprime(p+1)); forprime(r=nextprime(q+1), oo, my(c=0); for(k=1, oo, if(isprime(p^k + q^k - r^k), c+=1, break)); if(c==n, return(p)); p = q; q = r); \\ Daniel Suteu, Jan 04 2023

a(6) from Michael S. Branicky, Nov 29 2022

a(7) from Daniel Suteu, Jan 04 2023

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A358742 First of three consecutive primes p,q,r such that p^3 + q^3 - r^3 is prime.

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A358744 First of three consecutive primes p, q, r such that p + q - r, p^2 + q^2 - r^2 and p^3 + q^3 - r^3 are all prime.

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A358745 a(n) is the least prime p that is the first of three consecutive primes p, q, r such that p^i + q^i - r^i is prime for i from 1 to n but not n+1.

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