A380269 The minimal rank of an n-universal Z-lattice.

4, 5, 6, 7, 8, 13, 15, 16, 28, 30

Offset: 1

Author

Robin Visser, Jan 18 2025

Keywords

• nonn
• hard
• more

Comments

a(n) is the least positive integer k such that there exists a positive definite Z-lattice of rank k which represents all positive definite Z-lattices of rank n.

Byeong-Kweon Oh gives the lower bound a(24) >= 6673.

Examples

			If n <= 5, then the diagonal lattice I_{n+3} is an n-universal Z-lattice of minimal rank, thus a(n) = n+3 for all n <= 5.
		

Links

• Daejun Kim and Byeong-Kweon Oh, Representations of finite number of quadratic forms with same rank, Ramanujan J., 56 (2021), no. 2, 631-644.
• Byeong-Kweon Oh, Universal Z-lattices of minimal rank, Proc. Amer. Math. Soc., 128 (2000), no. 3, 683-689.
• O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag Berlin Heidelberg, 1973.

Crossrefs

Cf. A054911.