login
A176838
Primes p such that p^3 = q//3 for a prime q, where "//" denotes concatenation.
2
17, 157, 257, 277, 397, 677, 877, 997, 1217, 1697, 1997, 2417, 2777, 3257, 3517, 3697, 4157, 4177, 5077, 5197, 5897, 6277, 7417, 7517, 8377, 9397, 9497, 9677, 9857, 11197, 11597, 12157, 12457, 12697, 13397, 13477, 13877, 14057, 14197, 15017, 16477, 17597, 18097
OFFSET
1,1
COMMENTS
Diophantine equation p^3 = 10 * q + 3 with side condition p and q prime. Necessarily the LSD for such primes p is e = 7 and the two least significant digit strings are "17", "57", "77" or "97".
REFERENCES
J.-P. Allouche and J. Shallit, Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford University Press, 1980.
F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg - Berlin 1999.
LINKS
EXAMPLE
17^3 = 4913 = prime(94)//3, 17 = prime(7) is the first term.
157^3 = 3869893 = prime(32838)//3, 157 = prime(37) is the second term.
MAPLE
q:= n-> isprime(iquo(n^3, 10, 'd')) and d=3:
select(q, [ithprime(i)$i=1..3000])[]; # Alois P. Heinz, Sep 03 2021
MATHEMATICA
Select[Range[7, 20000, 10], PrimeQ[#]&&PrimeQ[FromDigits[Most[IntegerDigits[ #^3]]]]&] (* Harvey P. Dale, Oct 03 2013 *)
PROG
(Python)
from sympy import isprime, primerange
def ok(p): q, r = divmod(p**3, 10); return r == 3 and isprime(q)
print(list(filter(ok, primerange(2, 18200)))) # Michael S. Branicky, Aug 31 2021
(PARI) isok(p) = if (isprime(p), my(v=divrem(p^3, 10)); isprime(v[1]) && (v[2] == 3)); \\ Michel Marcus, Sep 03 2021
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2010
STATUS
approved