A030050 -id:A030050 - cp's OEIS frontend

A030051 Numbers from the 290-theorem.

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290

Offset: 1

The 290-theorem, conjectured by Conway and Schneeberger and proved by Bhargava and Hanke, asserts that a positive definite quadratic form represents all numbers iff it represents the numbers in this sequence. - T. D. Noe, Mar 30 2006

J. H. Conway and W. A. Schneeberger, personal communication.

Manjul Bhargava and Jonathan Hanke, Universal quadratic forms and the 290-Theorem Inventiones Math., 2005
Jangwon Ju and Byeong-Kweon Oh, Universal mixed sums of generalized 4- and 8-gonal numbers, arXiv:1809.03673 [math.NT], 2018. See p. 1.
Alexander J. Hahn, Quadratic Forms over Z from Diophantus to the 290 Theorem, Adv. Appl. Clifford Alg. 18 (2008), 665-676.
Jangwon Ju, Almost universal sums of triangular numbers with one exception, arXiv:2201.04355 [math.NT], 2022.
Yong Suk Moon, Universal Quadratic Forms and the 15-Theorem and 290-Theorem
K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.
Ivars Peterson, All Square: Science News Online
Ivars Peterson, MathTrek, All Square

Cf. A030050, A116582, A154363.

A116582 Numbers from Bhargava's 33 theorem.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 33

Offset: 1

Roger L. Bagula, Mar 23 2006

Bhargava's 33 theorem asserts that an integral quadratic form represents all odd numbers iff it represents 1, 3, 5, 7, 11, 15 and 33. - T. D. Noe, Mar 30 2006

Ivars Peterson, All Square: Science News Online

Cf. A030050 (numbers from the 15 theorem), A030051 (numbers from the 290 theorem), A154363.

Edited by N. J. A. Sloane, Apr 05 2006

More terms from T. D. Noe, Mar 30 2006

A154363 Numbers from Bhargava's prime-universality criterion theorem.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73

Offset: 1

Scott Duke Kominers (kominers(AT)fas.harvard.edu), Jan 07 2009

Bhargava's prime-universality criterion theorem asserts that an integer-matrix quadratic form represents all prime numbers if and only if it represents all numbers in this sequence.

H. Cohen, Number Theory, Springer, 2007, page 313.
M.-H. Kim, Recent developments on universal forms, Contemporary Math., 344 (2004), 215-228.

A030050 (numbers from the 15 theorem), A030051 (numbers from the 290 theorem), A116582 (numbers from the 33 theorem)

A158439 Numbers arising in Kane's conjecture on representing sets with sums of triangular numbers.

Original entry on oeis.org

1, 5, 7, 9, 11, 13, 17, 19, 25, 29, 35, 49, 89

Offset: 1

Jonathan Vos Post, Mar 19 2009

Jangwon Ju, Even universal sums of triangular numbers, arXiv:2408.11310 [math.NT], 2024.
Ben Kane, Representing Sets with Sums of Triangular Numbers, arXiv:0903.3026 [math.NT], Mar 17 2009.

Cf. A000217, A030050.

A298705 Numbers from the 15-theorem for universal Hermitian lattices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 14, 15

Offset: 1

Ralf Steiner, Jan 29 2018

Unlike the fifteen theorem for universal quadratic forms (A030050), this theorem includes the number 13 arising from the case Q(sqrt(-39)).

Byeong Moon Kim, Ji Young Kim, and Poo-Sung Park, The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields, Math. Comp. 79 (2010), 1123-1144.

Cf. A030050, A183071.

A304206 Numbers from the "octagonal theorem of sixty".

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 12, 13, 14, 18, 60

Offset: 1

Vincenzo Librandi, May 11 2018

Jangwon Ju, Universal sums of generalized pentagonal numbers, arXiv:1805.03434 [math.NT], 2018 (see page 2).
Jangwon Ju and Byeong-Kweon Oh, Universal sums of generalized pentagonal numbers, arXiv:1704.08826 [math.NT], 2017 (see page 6, Theorem 3.1).

Cf. A030050, A030051, A116582, A154363, A304204.

cp's OEIS Frontend

A030051 Numbers from the 290-theorem.

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A116582 Numbers from Bhargava's 33 theorem.

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A154363 Numbers from Bhargava's prime-universality criterion theorem.

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A158439 Numbers arising in Kane's conjecture on representing sets with sums of triangular numbers.

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A298705 Numbers from the 15-theorem for universal Hermitian lattices.

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A304206 Numbers from the "octagonal theorem of sixty".

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