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A053289
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First differences of consecutive perfect powers (A001597).
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16
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3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100
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OFFSET
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1,1
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COMMENTS
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Michel Waldschmidt writes: Conjecture 1.3 (Pillai). Let k be a positive integer. The equation x^p - y^q = k where the unknowns x, y, p and q take integer values, all >= 2, has only finitely many solutions (x,y,p,q). This means that in the increasing sequence of perfect powers [A001597] the difference between two consecutive terms [the present sequence] tends to infinity. It is not even known whether for, say, k=2, Pillai's equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers. See Sierpiński [1970], problem 238a, p. 116. - Jonathan Vos Post, Feb 18 2008
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REFERENCES
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Wacław Sierpiński, 250 problems in elementary number theory, Modern Analytic and Computational Methods in Science and Mathematics, No. 26, American Elsevier, Warsaw, 1970, pp. 21, 115-116.
S. S. Pillai, On the equation 2^x - 3^y = 2^X - 3^Y, Bull, Calcutta Math. Soc. 37 (1945) 15-20.
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LINKS
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FORMULA
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Formulas from Jakimczuk (2016):
Lim sup_{n->oo} a(n)/(2*n) = 1.
Lim inf_{n->oo} a(n)/(2*n)^(2/3 + eps) = 0. (End)
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EXAMPLE
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Consecutive perfect powers are A001597(14) = 121, A001597(13) = 100, so a(13) = 121 - 100 = 21.
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MATHEMATICA
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PROG
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(Python)
from sympy import mobius, integer_nthroot
if n==1: return 3
def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
kmin, kmax = 1, 2
while f(kmax)+1 >= kmax:
kmax <<= 1
rmin, rmax = 1, kmax
while True:
kmid = kmax+kmin>>1
if f(kmid)+1 < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
while True:
rmid = rmax+rmin>>1
if f(rmid) < rmid:
rmax = rmid
else:
rmin = rmid
if rmax-rmin <= 1:
break
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CROSSREFS
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Cf. A053707 (first differences of consecutive perfect prime powers).
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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