Yasutsugu Fujita and Noriaki Kimura

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 16, 2010, Number 2, Pages 16–23

**Full paper (PDF, 196 Kb)**

## Details

### Authors and affiliations

Yasutsugu Fujita

*Department of Mathematics, College of Industrial Technology
Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan*

Noriaki Kimura

*Department of Mathematics, College of Industrial Technology
Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan*

### Abstract

Let *f*(*X*_{1}, …, *X _{m}*) be a quadratic form in m variables

*X*

_{1}, …,

*X*with integer coefficients. Then it is well-known that the Diophantine equation

_{m}*f*(

*X*

_{1}, …,

*X*) = 0 has a nontrivial solution in integers if and only if the equation has a nontrivial solution in real numbers and the congruence

_{m}*f*(

*X*

_{1}, …,

*X*) ≡ 0 (mod

_{m}*N*) has a nontrivial solution for every integer

*N*> 1. Such a principle is called the Hasse principle. In this paper, we explicitly give several types of families of the Diophantine equations of degree two, not homogeneous, for which the Hasse principle fails.

### Keywords

- Hasse principle
- Diophantine equations
- Congruences

### AMS Classification

- 11D09
- 11A07

### References

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## Related papers

## Cite this paper

Fujita, Y., & Kimura, N. (2010). Infinitely many insolvable Diophantine equations. II. *Notes on Number Theory and Discrete Mathematics*, 16(2), 16-23.