A050791 -id:A050791 - cp's OEIS frontend

A050792 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z (see A050791). Sequence gives values of x.

9, 64, 73, 135, 334, 244, 368, 1033, 1010, 577, 3097, 3753, 1126, 4083, 5856, 3987, 1945, 11161, 13294, 3088, 10876, 16617, 4609, 27238, 5700, 27784, 11767, 26914, 38305, 6562, 49193, 27835, 35131, 7364, 65601, 50313, 9001, 11980, 39892, 20848

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

"One of the simplest cubic Diophantine equations is known to have an infinite number of solutions (Lehmer, 1956; Payne and Vaserstein, 1991). Any number of solutions to the equation x^3 + y^3 + z^3 = 1 can be produced through the use of the algebraic identity (9t^3+1)^3 + (9t^4)^3 + (-9t^4-3t)^3 = 1 by substituting in values of t. ...
"Although these are certainly solutions, the identity generates only one family of solutions. Other solutions such as (94, 64, -103), (235, 135, -249), (438, 334, -495), ... can be found. What is not known is if it is possible to parameterize all solutions for this equation. Put another way, are there an infinite number of families of solutions? Probable yes, but that too remains to be shown." [Herkommer]
Values of x associated with A050794.

			577^3 + 2304^3 = 2316^3 + 1.

Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, NY, 1999, page 370.
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Lewis Mammel, Table of n, a(n) for n = 1..368
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050791, A050793, A050794.

More terms from Michel ten Voorde.
Extended through 26914 by Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, May 08 2007

A050793 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1A050791), and increasing values of y in case of ties. Sequence gives values of y.

Original entry on oeis.org

10, 94, 144, 235, 438, 729, 1537, 1738, 1897, 2304, 3518, 4528, 5625, 8343, 9036, 9735, 11664, 11468, 19386, 21609, 31180, 35442, 36864, 33412, 38782, 35385, 41167, 44521, 51762, 59049, 50920, 72629, 76903, 83692, 67402, 80020, 90000

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Values of y associated with A050794.

			For the 10th term where y is 2304, 577^3 + 2304^3 = 2316^3 + 1.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Lewis Mammel, Table of n, a(n) for n = 1..368
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Cf. A050791, A050792, A050794.

More terms from Michel ten Voorde; no more with z<8192.
Extended through 44521 by Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, May 08 2007

A050794 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z (see A050791). Sequence gives values of x^3 + y^3 = z^3 + 1. For corresponding values of x, y, z see A050792, A050793, A050791 respectively.

Original entry on oeis.org

1729, 1092728, 3375001, 15438250, 121287376, 401947273, 3680797185, 6352182209, 7856862273, 12422690497, 73244501505, 145697644729, 179406144001, 648787169394, 938601300672, 985966166178, 1594232306569

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 28 2009

			577^3 + 2304^3 = 2316^3 + 1 = 12422690497.

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 "1729", p. 153.

Uwe Hollerbach and David Rabahy, Table of n, a(n) for n = 1..368, a(n) for n = 75..368 by David Rabahy, Oct 13 2015.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers.

Cf. A050791, A050792, A050793, A259753.

Extended through 1594232306569 by Jud McCranie, Dec 25 2000

A145383 Indices in A050791 for the subsequence A141326.

Original entry on oeis.org

1, 3, 6, 10, 13, 17, 20, 23, 30, 37, 38, 42, 44, 48, 51, 55, 58, 63, 68, 72, 73, 77, 83, 84, 87, 90, 94, 97, 99, 102, 106, 108, 111, 115, 121, 123, 125, 130, 133, 134, 136, 140, 142, 146, 150, 153, 156, 159, 164, 167, 169, 172

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Oct 10 2008

The first difference of this sequence is used to define the sequence A145384, which is of particular interest.

Lewis Mammel, Table of n, a(n) for n = 1..122

Cf. A050791, values of z where 1+z^3 = x^3 + y^3.
Cf. A141326, a subsequence of A050791 defined by a simple formula.
Cf. A145384, a sequence based on the first difference of this sequence.

The defining equation is A050791(a(n)) = A141326(n).

A145384 The number of terms of A050791 bracketed by successive terms of A141326.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 2, 6, 6, 0, 3, 1, 3, 2, 3, 2, 4, 4, 3, 0, 3, 5, 0, 2, 2, 3, 2, 1, 2, 3, 1, 2, 3, 5, 1, 1, 4, 2, 0, 1, 3, 1, 3, 3, 2, 2, 2, 4, 2, 1, 2, 4, 2, 0, 1, 2, 3, 1, 1, 1, 3, 0, 3, 1, 0, 3, 1, 1, 4, 2, 2, 1, 3, 3, 1, 2, 0, 3, 2, 5, 1, 1, 3, 6, 2, 4, 1, 0, 5, 2, 2, 2, 2, 3, 2, 3, 3, 0, 1

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Oct 10 2008

nonn

A141326 is a simply generated subsequence of A050791 and by observation it forms a natural measure of the parent sequence. The first several hundred terms of the parent sequence not belonging to A141326 are bracketed into groups with a small integral number of terms ( including 0 ) by the successive terms of the subsequence, A141326.
a(107),a(108) are the first occurrence of 2 consecutive 0's and a(119),a(120),a(121) are the first occurrence of 3 consecutive 0's. This leads to the following conjecture:
 -> 0 as n ->inf
where  = ( sum m=1,n of a(m) )/n

			0 = number of terms of A050791 preceding the first term of A141326
1 = number of terms of A050791 between the first and 2nd terms of A141326
2 = number of terms of A050791 between the 2nd and 3rd terms of A141326

Lewis Mammel, Table of n, a(n) for n = 1..122

Cf. A145383, A141326, A050791

a(1) = A145383(1) - 1

a(n) = A145383(n) - A145383(n-1) - 1 ; n>1

A143936 Subsequence of A050791, "Fermat near misses", generated by iteration of a linear form derived from Ramanujan's parametric formula for equal sums of two pairs of cubes.

Original entry on oeis.org

5262, 2262756, 972979926, 418379105532, 179902042398942, 77357459852439636

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008

nonn

The formulas give an approximately geometric progression of values, z, such that 1 + z^3 = x^3 + y^3, along with the values for x and y. Iteration yields large values of x,y and z presumably unobtainable by exhaustive search.

			1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777

Charles Edward Sandifer, The Early Mathematics of Leonhard Euler, 2007, pp. 102-103.

Wolfram Mathworld, Diophantine Equation 3rd Powers

Cf. A050791, A141326.

/*
File: form.bc
Usage: bc form.bc
( In UNIX shell, e.g. bash on Cygwin )
*/
define a(x){ return( 321*x^2 + 216*x + 36 ); }
define b(x){ return( sqrt(a(x)) ); }
define n(z){ auto a,x; x=3; a = 215*z+12*b(z)+72 ;
a;b(a); return(v(a)); }
define v(z){ auto a,b,x,y,i,j,k,l;
a = z; b = ( a + b(a) )/2;
a = -a; x=3; y = 1-a*x;
i=a*x+y; j=b+x^2*y; k=b*x+y; l=a+x^2*y;
-a; b; i;j;k;l; i^3+j^3; k^3+l^3;
return ( -a ); }
z=144; v(z) ; z=n(z); z=n(z); z=n(z); /* ... etc. */

In Ramanujan's parametric formula:
(a*x+y)^3 + (b+x^2*y)^3 = (b*x+y)^3 + (a+x^2*y)^3
with
a^2 + a*b + b^2 = x*y^2,
we set x=3, ax+y=1 and obtain a quadratic equation for b in terms of a
( Since 'a' is always negative we write it explicitly as '-a' and solve for positive 'a' )
The surd of the quadratic formula then becomes:
sqrt(321*a^2 + 216*a + 36)
and we require that this be an integer. After finding an initial value of 'a' which satisfies this condition by inspection of the sequence A050791, we use Euler's method to find the bilinear recursion: ( with s_i == sqrt(321*a_i^2 + 216*a_i + 36) )
a_i+1 = 215*a_i + 12*s_i + 72
s_i+1 = 215*s_i + 3852*a_i + 1296
and these yield the values of x,y and z from Ramanujan's formula.

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.

Original entry on oeis.org

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

A141326 Subsequence of 'Fermat near misses' which is generated by a simple formula based on the cubic binomial expansion along with formulas for the corresponding terms in the expression, x^3 + y^3 = z^3 + 1.

Original entry on oeis.org

12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, 131802, 186660, 257088, 345786, 455670, 589872, 751740, 944838, 1172946, 1440060, 1750392, 2108370, 2518638, 2986056, 3515700, 4112862, 4783050, 5531988, 6365616, 7290090, 8311782, 9437280, 10673388

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Aug 03 2008

From Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008: (Start)
In Ramanujan's parametric equation: (ax+y)^3 + (b+x^2y)^3 = (bx+y)^3 + (a+x^2y)^3
where a^2 + ab + b^2 = 3xy^2.
This sequence is obtained by setting a=0, y=1 and finding the solution to b^2=3x:
b=3n, x=3n^2. (End)

			For a(1)=12: 1 + 12^3 = 9^3 + 10^3 = 1729.

Colin Barker, Table of n, a(n) for n = 1..1000
Tito Piezas III and Eric Weisstein, Diophantine Equation--3rd Powers [Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A050791, A050792, A050793, A050794.

Vec(6*x*(2 + 15*x + 18*x^2 + x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Oct 26 2019

a(n) = 9*n^4 + 3*n, with b(n) = 9*n^4 and c(n) = 9*n^3 + 1 we have 1 + a(n)^3 = b(n)^3 + c(n)^3.
From Colin Barker, Oct 25 2019: (Start)
G.f.: 6*x*(2 + 15*x + 18*x^2 + x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
a(n) = 3*(n + 3*n^4).
(End)

Edited by Joerg Arndt, Oct 26 2019

A229383 Near-miss counterexample to Fermat's Last Theorem for exponent 3.

Original entry on oeis.org

252688188876561, 363025382900625, 399951361478656

Offset: 1

Joe Sondow and Jonathan Sondow, Sep 24 2013

In the Fermat "equation" 252688188876561^3 + 363025382900625^3 = 399951361478656^3 the left side is 0.000000002% larger than the right side. However, the ones digit on both sides is the same, namely, 6.

4th powers of A229382, David S. Cohen's near-miss counterexample to Fermat's Last Theorem for exponent 12.

S. Singh, The Simpsons and Their Mathematical Secrets, Bloomsbury USA, 2013.

C. Goff, Animating Popular Mathematics: "The Simpsons and Their Mathematical Secrets", AMS Notices, 62 (2015), 40-44.
R. Perrin, Sur l’intégration indéterminée x^3 + y^3 = z^3, Bulletin de la S. M. F., tome 13 (1885), pp. 194-197.

Cf. A050791, A229382.

a(n) = A229382(n)^4.

A160054 Primes prime(k) such that prime(k)^2 + prime(k+1)^2 - 1 is a perfect square.

Original entry on oeis.org

7, 11, 23, 109, 211, 307, 1021, 4583, 42967, 297779, 1022443, 1459811, 10781809, 125211211, 11673806759, 3019843939831, 40047392632801, 88212019638251209, 444190204424015227, 57852556614292865039, 9801250757169593701501, 64747502900142088755541, 619216322498658374863033

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 01 2009

nonn

An infinite number of solutions exists for a^2 + b^2 - 1 = c^2 over the set of natural numbers a, b, c.
If we constrain these to b=a+2, i.e., 2a^2 + 4a + 3 = c^2, the solutions are with a = 1, 11, 69, 407, 2377, ... (The twin prime 11 is also in this sequence here. The solutions can be generated recursively from a(0)=1, m(0)=3 and a(k+1) = 3*a(k) + 2*m(k) + 2, m(k+1) = 4*a(k) + 3*m(k) + 4.)
Filtering these solutions for prime pairs a(k) and b(k) would generate the subset of lower twin primes in the sequence.
The equivalent procedure can be carried out for other prime gaps 2*d such that prime(k)=a, prime(k+1)=a+2*d, 2*a^2 + 4*a*d + 4*d^2 - 1 = m^2. This decomposes the sequence into classes according to the gap 2*d.
a(17) > 5*10^12. - Donovan Johnson, May 17 2010

			7^2 + 11^2 - 1 = 13^2.
11^2 + 13^2 - 1 = 17^2.
23^2 + 29^2 - 1 = 37^2.
109^2 + 113^2 - 1 = 157^2.
211^2 + 223^2 - 1 = 307^2.
307^2 + 311^2 - 1 = 19^2*23^2.
1021^2 + 1031^2 - 1 = 1451^2.
4583^2 + 4591^2 - 1 = 13^2*499^2.

Cf. A050791, A129288, A271050.

[n: n in [0..2*10^7] | IsSquare(n^2+NextPrime(n+1)^2-1) and IsPrime(n)]; // Vincenzo Librandi, Aug 02 2015

lst = {}; p = q = 2; While[p < 4000000000, q = NextPrime@ p; If[ IntegerQ[ Sqrt[p^2 + q^2 - 1]], AppendTo[lst, p]; Print@ p]; p = q]; lst (* Robert G. Wilson v, May 31 2009 *)

p=2;forprime(q=3,1e6,if(issquare(q^2+p^2-1),print1(p", "));p=q) \\ Charles R Greathouse IV, Nov 06 2014

is(n)=issquare(n^2+nextprime(n+1)^2-1)&&isprime(n) \\ Charles R Greathouse IV, Nov 29 2014

{A000040(k): A069484(k)-1 in A000290}.

Edited and 4 more terms from R. J. Mathar, May 08 2009
a(13) from Robert G. Wilson v, May 31 2009
a(15)-a(16) from Donovan Johnson, May 17 2010
More terms from Jinyuan Wang, Jan 09 2021

cp's OEIS Frontend

A050792 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z (see A050791). Sequence gives values of x.

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A050793 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1A050791), and increasing values of y in case of ties. Sequence gives values of y.

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A050794 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z (see A050791). Sequence gives values of x^3 + y^3 = z^3 + 1. For corresponding values of x, y, z see A050792, A050793, A050791 respectively.

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A145383 Indices in A050791 for the subsequence A141326.

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A145384 The number of terms of A050791 bracketed by successive terms of A141326.

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A143936 Subsequence of A050791, "Fermat near misses", generated by iteration of a linear form derived from Ramanujan's parametric formula for equal sums of two pairs of cubes.

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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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A141326 Subsequence of 'Fermat near misses' which is generated by a simple formula based on the cubic binomial expansion along with formulas for the corresponding terms in the expression, x^3 + y^3 = z^3 + 1.

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