Thue's Lemma on "Small" Solutions to Linear Congruences

Explanations 7

• Thue's lemma is a simple lemma which essentially shows that there exists a small solution to a two-variable linear congruence, with many applications as we will see... Essential…

  By Vincent Huang, Victor Wang

• ppp is a prime and kkk is an integer so that ppp divides k2+5k^2+5k2+5. Prove that there are integers x,yx,yx,y with p2=x2+5y2p^2=x^2+5y^2p2=x2+5y2.

  By amaterasu

• Let ppp be a prime for which there exists a positive integer aaa such that ppp divides 2a2−12a^2-12a2−1. Prove that, there exist integers bbb and ccc so that, p=2b2−c2p=2b^2-c^2…

  By amaterasu

• If ppp is a prime of the form 3k+13k+13k+1, prove there are integers a,ba,ba,b such that p=a2+ab+b2p=a^2+ab+b^2p=a2+ab+b2.

  By amaterasu

• Let n∈{−1,−2,−3}n\in\{-1,-2,-3\}n∈{−1,−2,−3}. If nnn is a quadratic residue of a prime ppp, then there are integers a,ba,ba,b so that, a2−nb2=pa^2-nb^2=pa2−nb2=p.

  By amaterasu

• Two Dimensional version. Let n≥2,r=nn\geq2,r=\sqrt{n}n≥2,r=n​ and a,b,c,da,b,c,da,b,c,d be arbitrary integers. There exist w,x,y,zw,x,y,zw,x,y,z with at least one of y,zy,zy,z n…

  By amaterasu

• Prove the generalization. Let α\alphaα and β\betaβ are two real numbers so that αβ≥p\alpha\beta \geq pαβ≥p. Then for an integer xxx, there are integers a,ba,ba,b with 0<∣a∣<α0<|…

  By amaterasu