What Are Diophantine Equations?
Read this topicExplanations 5
- Square+Cube=Square!
By Kalpok Guha
- Why is 2\sqrt{2}2 irrational? Suppose for the sake of contradiction that a2=2a^2=2a2=2 has a rational solution a=pqa=\frac{p}{q}a=qp. Let gcd(p,q)=1\gcd(p,q)=1gcd(p,q)=1, so …
By Jason Chen, Alex Federspiel, Victor Wang
- Linear Diophantine Equations Let a,b∈Za, b\in{Z}a,b∈Z. The equation ax+by=cax + by = cax+by=c has a solution (x,y)∈Z2(x,y) \in{Z^2}(x,y)∈Z2 if and only if ccc is a multiple of t…
By Vinitha Ranganeni, Victor Wang
- Warm-Up You and your best friend Diophantus (who doesn't have a friend named Diophantus?) are in a candy store. You are entirely captivated by two types of candy - chocolates an…
By Catherine Yeo
- Fermat's Last Theorem Around 1637, Fermat famously wrote It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power h…
By Victor Wang, Victor Wang
Problems 11
- How can you tell if a Diophantine equation has integer solutions?
- Solve for the possible integer solution(s) for this equation. x2+xy+y2=(x+y3+1)3.x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.x2+xy+y2=(3x+y+1)3.
- Find all solutions to (m2+n)(m+n2)=(m−n)3(m^2+n)(m + n^2)= (m - n)^3(m2+n)(m+n2)=(m−n)3, where mmm and nnn are non-zero integers.
- Find all integer solutions (x,y)(x,y)(x,y) to 7x2−13xy+7y23=∣x−y∣+1.\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.37x2−13xy+7y2=∣x−y∣+1.
- (x2−y2)2=16y+1(x^2-y^2)^2=16y+1(x2−y2)2=16y+1, x,y are integers. What are the possible (x,y) (type as (x,y))
- Which of these is an integer solution to the equation 4x3y=x2y2−5x+y+14x^3 y=x^2 y^2-5x+y+14x3y=x2y2−5x+y+1.
- If p+1=2x2p+1=2x^2p+1=2x2, p2+1=2y2p^2+1=2y^2p2+1=2y2, and ppp is a prime number, what is (p,x,y)(p,x,y)(p,x,y)?
- Are there any integer solutions to the equation 16x+48y=27616x+48y=27616x+48y=276?
- Determine if the following Diophantine equation has any solutions: 5x−7y=−1055x-7y=-1055x−7y=−105
- এই সমীকরণটির সকল পূর্ণ সংখ্যার সমাধান বের কর। x2+xy+y2=(x+y3+1)3.x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.x2+xy+y2=(3x+y+1)3.
বাংলা
- How have we used integer solutions to Fermat's equation in real life? Hint: they shaped history.