A166929 -id:A166929 - cp's OEIS frontend

A166930 Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some positive coprime integers a, b, c.

2165017, 15512114571284835412957, 368440923990671763222767414151367493861848396861, 29032470413228645503712143213832535500985227130245791625262982715784415755764157625

Offset: 1

Max Alekseyev, Oct 23 2009

nonn

Square roots of the hypotenuses of Pythagorean triangles in which the hypotenuse and the sum of the legs are squares. In a letter to Mersenne in the year 1643, Fermat asserted that the smallest such triangle has the legs 4565486027761 and 1061652293520, and the hypotenuse a(1)^2 = 4687298610289.
Subsequence of A166929 which allows a,b be nonzero.
Values of m in coprime solutions to 2m^4 = c^4 + d^2 with d < c^2 (so that a,b = (c^2 +- d)/2). Corresponding values of c are given in A167438.

W. Sierpinski. Pythagorean Triangles. Dover Publications, 2003, ISBN 0-486-43278-5.

Gerry Martens, Table of n, a(n) for n = 1..13

Cf. A166929, A167437, A167438.

Edited by Max Alekseyev, Nov 03 2009

A167437 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime integers a, b, m.

Original entry on oeis.org

1, 1343, 2372159, 9788425919, 5705771236038721, 17999572487701067948161, 173658539553825212149513251457, 75727152767742719949099952561135816319, 437825148963391521638828389137484882137402791039

Offset: 1

Max Alekseyev, Nov 03 2009

nonn

Corresponding values of m are given in A166929 - see it for further details.
Terms with positive a, b are given in A167438.
This is the absolute value of a bisection of a generalized Somos-5 sequence. - Michael Somos, Nov 04 2022

{a(n) = local(x, v); n = abs(2*n+1); (x = m-> v[abs(m)+1]); v = vector(max(3, n+1), m, 1); v[3] = -3; for(k=3, n, v[k+1] = -(13*x(k-1)*x(k-4) + 42*x(k-2)*x(k-3)) / x(k-5)); abs(x(n))}; /* Michael Somos, Nov 04 2022 */

A167438 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime positive integers a, b, m.

Original entry on oeis.org

2372159, 17999572487701067948161, 437825148963391521638828389137484882137402791039, 34168080993535113552180464917346868292958739991398355562578195440360113112814117057

Offset: 1

Max Alekseyev, Nov 03 2009

nonn

Corresponding values of m are given in A166930.
Subsequence of A167437 which does not restrict a, b to positive integers.
Terms 5..8 found. - Gerry Martens, Jan 15 2019

Gerry Martens, Table of n, a(n) for n = 1..13

Cf. A166929, A166930, A167437.

A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)a(n-4) + 42a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 1, 3, 13, 113, 1525, 57123, 2165017, 262621633, 42422452969, 14070212996451, 7658246457672229, 10650393355715621873, 15512114571284835412957, 75606222210863532170808003, 452005526897888844293504165425

Offset: 0

Michael Somos, Jan 29 2023

nonn

The elliptic curve y^2 = x^3 - 2*x (LMFDB label 256.b1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). The x and y coordinates of n*P + T have denominators a(n)^2 and a(n)^3 respectively.

			2*P + T = (-8/9, -28/27). 3*P + T  = (-1/169, 239/2197).

LMFDB, Elliptic Curve 256.b1 (Cremona label 256b1)

Cf. A166929.

a[ m_] := With[{n = Abs[m]}, If[ n<3, {1, 1, 3}[[n+1]], (-a[n-1]*a[n-4] + 42*a[n-2]*a[n-3])/a[n-5]]];

{a(n) = my(E = ellinit([-2, 0])); sqrtint(denominator(elladd(E, [0, 0], ellmul(E, [2, 2], n))[1]))};

{a(n) = my(A); n = abs(n); A = vector(max(4, n+1), k, 1); A[3] = 3; A[4] = 13; for(k = 4, n, A[k+1] = (if(k%2, 4, 8)*A[k]*A[k-2] + A[k-1]^2)/A[k-3]); A[n+1]};

a(2*n-1) = A166929(n) for all n in Z.

cp's OEIS Frontend

A166930 Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some positive coprime integers a, b, c.

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A167437 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime integers a, b, m.

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A167438 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime positive integers a, b, m.

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Author

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A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)a(n-4) + 42a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

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cp's OEIS Frontend

A166930 Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some positive coprime integers a, b, c.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A167437 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime integers a, b, m.

Views

Author

Keywords

Comments

Programs

PARI

A167438 Positive integers c such that c^2 = a + b and a^2 + b^2 = m^4 for some coprime positive integers a, b, m.

Views

Author

Keywords

Comments

Links

Crossrefs

A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)*a(n-4) + 42*a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A360187 Generalized Somos-5 sequence with a(n) = (-a(n-1)a(n-4) + 42a(n-2)*a(n-3))/a(n-5), a(-n) = a(n), a(0) = a(1) = 1, a(2) = 3.