A098970 -id:A098970 - cp's OEIS frontend

A099807 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'b' to be 1 mod 12. This is the sequence of 'b' values, sorted by the magnitude of c.

37, 2137, 8929, 1801, 48817, 6637, 57241, 133597, 151477, 334717, 3889, 127717, 786697, 735781, 1154017, 38557, 1662229, 2446777, 3882661, 3811669, 2747449, 3716701, 5634637, 3600097, 9836221, 10591849, 7139569, 9473161, 11395309

Offset: 0

James R. Buddenhagen, Oct 26 2004

nonn

All terms of this sequence are of the form -3*M^4+N^4+6*M^2*N^2 for some pair M,N of relatively prime positive integers of opposite parity. For each n, a=A099806[n], b=A099807[n] are prime numbers and a^3 + b^3 = c^2, for some integer c. c is divisible by 12 and A098970 gives the values of c/12.

			37 is in the sequence because 37 is a prime congruent to 1 mod 12 and 11^3+37^3=228^2.

James Buddenhagen, Two Primes Cubed which Sum to a Square.

Cf. A099806, A098970, A099808, A099809.

A099808 If a,b are primes which satisfy the Diophantine equation a^3 + b^3 = c^2, then this sequence consists of the numbers sqrt((a+b)/48), sorted by the magnitude of c.

Original entry on oeis.org

1, 15, 28, 35, 44, 44, 55, 56, 91, 90, 88, 119, 161, 165, 200, 184, 273, 319, 285, 357, 377, 400, 380, 434, 550, 517, 592, 615, 638, 667, 682, 666, 740, 697, 784, 688, 825, 682, 846, 770, 893, 814, 868, 925, 775, 899, 885, 1007, 1045, 1040, 1078, 1184, 1015

Offset: 0

James R. Buddenhagen, Oct 26 2004

For each n let a=A099806(n), b=A099807(n). Then sqrt((a+b)/48) is an integer and equals A099808(n). Note that a^3 + b^3 = c^2 factors as (a+b)*(a^2-a*b+b^2). The first factor (a+b) is 48*d^2, some d. This sequence tabulates the d values. Remember, a and b are prime numbers.

			From 11^3 + 37^3 = 228^2 we get sqrt((a+b)/48) = (11+37)/48 = 1, so 1 is in the sequence. [corrected by _Harvey P. Dale_, Apr 12 2011]

James Buddenhagen, Two Primes Cubed which Sum to a Square.

Cf. A099806, A099807, A098970, A099809.

A099806 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'a' to be -1 mod 12. This is the sequence of 'a' values, sorted by the magnitude of c.

Original entry on oeis.org

11, 8663, 28703, 56999, 44111, 86291, 87959, 16931, 246011, 54083, 367823, 552011, 457511, 571019, 765983, 1586531, 1915163, 2437751, 16139, 2305883, 4074743, 3963299, 1296563, 5440991, 4683779, 2238023, 9682703, 8681639, 8142803

Offset: 0

James R. Buddenhagen, Oct 26 2004

nonn

All terms of this sequence are of the form 3*M^4-N^4+6*M^2*N^2 for some pair M,N of relatively prime positive integers of opposite parity.

			11 is in the sequence because 11 is -1 mod 12 and 11^3+37^3 = 228^2.

James Buddenhagen, Two Primes Cubed which Sum to a Square.

Cf. A099807, A098970, A099808, A099809.

A099809 Let a,b be prime numbers satisfying the Diophantine equation a^3+b^3=(a+b)(a^2-ab+b^2)=c^2. Then the second factor a^2-ab+b^2 is 3e^2 for some integer e. This sequence tabulates the 'e' values, sorted by magnitude of c.

Original entry on oeis.org

19, 4513, 14689, 32401, 26929, 48019, 44641, 72739, 124099, 179683, 211249, 288979, 395089, 386131, 587233, 905059, 1040419, 1410049, 2237011, 1919779, 2078209, 2220451, 2950963, 2767489, 4919971, 5582449, 5019889, 5255761

Offset: 0

James R. Buddenhagen, Oct 26 2004

nonn

For each n let a=A099806[n], b=A099807[n], c/12=A098970. Then a^3+b^3=c^2. The left side factors as (a+b)*(a^2-a*b+b^2). The second factor is 3*e^2 for some integer e. The sequence tabluates the 'e' values. These 'e' values all have the form 3*M^4+N^4, for some pair M,N of relatively prime integers of opposite parity. Remember, a and b are prime numbers.

			11^3+37^3=228^2, 11^2-11*37+37^2=3*e^2 with e=19, so 19 is in the sequence.

James Buddenhagen, Two Primes Cubed which Sum to a Square.

Cf. A099806, A099807, A098970, A099808.

cp's OEIS Frontend

A099807 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'b' to be 1 mod 12. This is the sequence of 'b' values, sorted by the magnitude of c.

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A099808 If a,b are primes which satisfy the Diophantine equation a^3 + b^3 = c^2, then this sequence consists of the numbers sqrt((a+b)/48), sorted by the magnitude of c.

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A099806 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'a' to be -1 mod 12. This is the sequence of 'a' values, sorted by the magnitude of c.

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Author

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Examples

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A099809 Let a,b be prime numbers satisfying the Diophantine equation a^3+b^3=(a+b)(a^2-ab+b^2)=c^2. Then the second factor a^2-ab+b^2 is 3e^2 for some integer e. This sequence tabulates the 'e' values, sorted by magnitude of c.

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Author

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Crossrefs

cp's OEIS Frontend

A099807 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'b' to be 1 mod 12. This is the sequence of 'b' values, sorted by the magnitude of c.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A099808 If a,b are primes which satisfy the Diophantine equation a^3 + b^3 = c^2, then this sequence consists of the numbers sqrt((a+b)/48), sorted by the magnitude of c.

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Author

Keywords

Comments

Examples

Links

Crossrefs

A099806 If a,b are prime numbers satisfying the Diophantine equation a^3+b^3=c^2, then a is -1 mod 12 and b is 1 mod 12, or vice versa. Choose 'a' to be -1 mod 12. This is the sequence of 'a' values, sorted by the magnitude of c.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A099809 Let a,b be prime numbers satisfying the Diophantine equation a^3+b^3=(a+b)*(a^2-a*b+b^2)=c^2. Then the second factor a^2-a*b+b^2 is 3*e^2 for some integer e. This sequence tabulates the 'e' values, sorted by magnitude of c.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A099809 Let a,b be prime numbers satisfying the Diophantine equation a^3+b^3=(a+b)(a^2-ab+b^2)=c^2. Then the second factor a^2-ab+b^2 is 3e^2 for some integer e. This sequence tabulates the 'e' values, sorted by magnitude of c.