A050788 -id:A050788 - cp's OEIS frontend

A050787 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (0 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of z.

9, 144, 172, 505, 577, 729, 904, 1010, 1210, 2304, 3097, 3753, 5625, 6081, 6756, 8703, 11664, 12884, 16849, 18649, 21609, 24987, 29737, 36864, 37513, 38134, 38239, 41545, 49461, 51762, 59049, 66465, 68010, 69709, 71852, 73627, 78529

Offset: 1

Patrick De Geest, Sep 15 1999

n^3 - 1 is expressible as the sum of two distinct positive cubes. [corrected by Altug Alkan, Apr 11 2016]
The subsequence of primes in the sequence begins: 577, 38239, 69709. - Jonathan Vos Post, May 13 2010
Sequence is infinite. One subsequence is b (m) = 9 m^4 = {9, 144, 729, 2304, 5625, 11664, 21609, 36864, 59049, ...} = a (1, 2, 6, 10, 13, 17, 21, 24, 31, ...). - Zak Seidov, Sep 16 2013

			2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.

Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..104 (first 60 terms from Alcover)
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050788, A050789, A050790, A050791.

N:= 10000: # to get all entries <= N
P:= proc(r)
  local dcands, xs;
  dcands:= select(d -> issqr(-3*d^4+12*d*r), numtheory[divisors](r));
  xs:= map(d -> [solve(d^2-3*d*x+3*x^2-r/d,x)], dcands);
  select(p -> p[1]<>p[2], select(type,xs,list(posint)));
end proc:
select(z -> nops(P(z^3-1))>0, [$1..N]); # Robert Israel, Jun 09 2014

r[z_] := Reduce[1 < x < y < z && x^3 + y^3 == z^3 - 1, {x, y}, Integers]; Reap[z = 4; While[z < 10^5, rz = r[z]; If[rz =!= False, Print[xyz = {x, y, z} /. ToRules[rz]]; Sow[xyz[[3]]]]; z++]][[2, 1]] (* Jean-François Alcover, Dec 27 2011, updated Feb 11 2014 *)

is(n)=if(n<2,return(0));my(c3=n^3);for(a=2,sqrtnint(c3-5,3),if(ispower(c3-1-a^3,3),return(1)));0 \\ Charles R Greathouse IV, Oct 26 2014

T=thueinit('x^3+1);
is(n)=n>8&&#select(v->min(v[1],v[2])>1,thue(T,n^3-1))>0 \\ Charles R Greathouse IV, Oct 26 2014

More terms from Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Definition corrected by Robert Israel, Jun 09 2014

A050789 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. The values of z (see A050787) are arranged in monotonically increasing order. Sequence gives values of y.

Original entry on oeis.org

8, 138, 138, 426, 486, 720, 823, 812, 1207, 2292, 2820, 3230, 5610, 5984, 6702, 8675, 11646, 11903, 16806, 17328, 21588, 24965, 27630, 36840, 31212, 37887, 33857, 34566, 49409, 46212, 59022, 66198, 66167, 56503, 69479, 64165, 78244, 89970

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			575^3 + 2292^3 = 2304^3 - 1.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.

Jean-François Alcover, Table of n, a(n) for n = 1..60
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050787, A050788, A050790.

More terms from Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, Feb 22 2009

A173515 Consider positive integer solutions to x^3 + y^3 = z^3 - n or 'Fermat near misses' of 1, 2, 3 ... Arrange known solutions by increasing values of n. Sequence gives value of lowest z for a given n.

Original entry on oeis.org

9, 7, 2, 812918, 18, 217, 4, 3, 9730705, 332, 14, 135, 3, 19, 156, 16, 15584139827, 3, 139643, 6, 1541, 4, 2220422932, 5, 14, 4, 445, 12205, 9, 8, 16234, 815, 31, 4

Offset: 1

Andy Martin, Feb 20 2010

The submitted values are for z when 0 < n < 51. There is no solution for any n congruent to 4 or 5 mod 9. This eliminates 4,5,13,14,22,23,31,32,40,41,49 and 50 in the 0-to-50 range.
Per the Elsenhans and Jahnel link there are no solutions found for 3, 33, 39 and 42 in the 0-to-50 range, with a search bound of 10^14.
If sequences could contain 'nil' for no solution, and '?' for cases where a solution is not known, but might exist, then a more concise definition is possible: Least positive integer such that a(n)^3 - n is the sum of two positive cubes. The sequence would then start: 9, 7, ?, nil, nil, 2.

			6^3 + 8^3 = 9^3 - 1: There are no solutions when n = 1 for z < 9, thus the first term is 9.
5^3 + 6^3 = 7^3 - 2: There are no solutions for z < 7, thus the second term is 7.
It is unknown if there is a solution when n = 3.
It is known there are no solutions when n = 4 and 5.
1^3 + 1^3 = 2^3 - 6, thus the third term is 2.

Andreas-Stephan Elsenhans and Joerg Jahnel, List of solutions of x^3 + y^3 + z^3 = n for n < 1000 neither a cube nor twice a cube
D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele, on Solving the Diophantine Equation x^3 + y^3 + z^3 = k on a Vector Computer
Kenji Koyama, Yukio Tsuruoka, and Hiroshi Sekigawa, On Searching For Solutions of the Diophantine Equation x^3 + y^3 + z^3 = n, Math. Comp. 66 (1997), 841-851.
Eric S. Rowland, Known Families of Integer Solutions of x^3 + y^3 + z^3 = n

Cf. A050788, A159935.

# x^3 + y^3 = z^3 - n
# Solve for all z less than z_limit, and
# n less than n_limit.
# When n = 7, z = 812918 and faster code and language are needed.
# However, by optimizing this code slightly and running for 2 days
# the author was able to search all z < 164000 and n < 100
#
n_limit = 7 # Configure as desired
z_limit = 20 # Configure as desired
h = {}
(2..z_limit).each{ |z|
  (1..(z-1)).each{ |y|
  (1..(y)).each{ |x|
  n = z*z*z - x*x*x - y*y*y
  if n > 0 && n < n_limit && h[n].nil?
  puts "Found z = #{z} when #{x}^^3 + #{y}^^3 = #{z}^^3 - #{n}"
  h[n] = z
  end
} } }
print "\nPartial sequence generated when n < #{n_limit} and z is searched to #{z_limit} is:\n"
h.sort.each{|k,v| print "#{v}, " }
print "\b\b \n"

Author conjectures that no explicit formula or recurrence exists.

A050790 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of x^3 + y^3 = z^3 - 1.

Original entry on oeis.org

728, 2985983, 5088447, 128787624, 192100032, 387420488, 738763263, 1030300999, 1771560999, 12230590463, 29704593672, 52861038776, 177978515624, 224866629440, 308367729215, 659184444926, 1586874322943

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			575^3 + 2292^3 = 2304^3 - 1 = 12230590463.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.

Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050787, A050788, A050789.

More terms from Jud McCranie, Dec 25 2000

cp's OEIS Frontend

A050787 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (0 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of z.

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A050789 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. The values of z (see A050787) are arranged in monotonically increasing order. Sequence gives values of y.

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A173515 Consider positive integer solutions to x^3 + y^3 = z^3 - n or 'Fermat near misses' of 1, 2, 3 ... Arrange known solutions by increasing values of n. Sequence gives value of lowest z for a given n.

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A050790 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of x^3 + y^3 = z^3 - 1.

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