Some universal quadratic sums over the integers

  • Received: 01 August 2019
  • Primary: 11E25; Secondary: 11D85, 11E20

  • Let $ a,b,c,d,e,f\in\mathbb N $ with $ a\geq c\geq e>0 $, $ b\leq a $ and $ b\equiv a\ ({\rm{mod}}\ 2) $, $ d\leq c $ and $ d\equiv c\ ({\rm{mod}}\ 2) $, $ f\leq e $ and $ f\equiv e\ ({\rm{mod}}\ 2) $. If any nonnegative integer can be written as $ x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 $ with $ x,y,z\in\mathbb Z $, then the ordered tuple $ (a,b,c,d,e,f) $ is said to be universal over $ \Bbb Z $. Recently, Z.-W. Sun found all candidates for such universal tuples over $ \Bbb Z $. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $ (a,b,c,d,e,f) $ in Sun's list of candidates are indeed universal over $ \mathbb Z $. For example, we prove the universality of $ (16,4,2,0,1,1) $ over $ \Bbb Z $ which is related to the form $ x^2+y^2+32z^2 $.

    Citation: Hai-Liang Wu, Zhi-Wei Sun. Some universal quadratic sums over the integers[J]. Electronic Research Archive, 2019, 27: 69-87. doi: 10.3934/era.2019010

    Related Papers:

  • Let $ a,b,c,d,e,f\in\mathbb N $ with $ a\geq c\geq e>0 $, $ b\leq a $ and $ b\equiv a\ ({\rm{mod}}\ 2) $, $ d\leq c $ and $ d\equiv c\ ({\rm{mod}}\ 2) $, $ f\leq e $ and $ f\equiv e\ ({\rm{mod}}\ 2) $. If any nonnegative integer can be written as $ x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 $ with $ x,y,z\in\mathbb Z $, then the ordered tuple $ (a,b,c,d,e,f) $ is said to be universal over $ \Bbb Z $. Recently, Z.-W. Sun found all candidates for such universal tuples over $ \Bbb Z $. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $ (a,b,c,d,e,f) $ in Sun's list of candidates are indeed universal over $ \mathbb Z $. For example, we prove the universality of $ (16,4,2,0,1,1) $ over $ \Bbb Z $ which is related to the form $ x^2+y^2+32z^2 $.



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