Perfect Squares,Cubes,Fourth Powers and More Many problems in number theory involve expressions with perfect squares,perfect cubes or perfect fourth powers.There are problems that require us tofind integer solutions to equations with unknown exponents.Below I will list veral common types of problems and strategies to tackle them.Of cour,the list is not exhaustive.
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1.Finding integer solutions to an equation involving perfect squares,cubes or
fourth powers.Many of the problems can be tackled by considering the pos-sible residues,modulo a suitable number.For perfect squares,trying considering modulo4,8,16,3,5,7,11or13.For perfect cubes,it may be uful to take modulo7,9or,ocassionally,modulo13.For perfect fourth powers,you might wish to consider modulo5or16.Of cour,the are just suggestions,and it really depends on the problem.
For equations that involve only perfect squares in two variables,it may be uful to consider the equation as a quadratic in one variable,and u the fact that discrim-inant is non-negative for a solution to exist,or that the discriminant is a perfect square(in order for the solutions to be an integer).This approach,however,has limited us and may sometimes lead to a dead end.
Another approach would be to try to factori the given expression to try to derive uful divisibility properties or inequalities that will allow you to reduce the scope of consideration to just a few cas.Fa
ctorisation is always one approach that you should keep in mind,as it may sometimes yield surprising results.
Alternatively,to show that a certain equation has no integer solutions,we can also u Fermat’s method of infinite descent.To show that an equation involving per-fect squares has infinitely many solutions,we may want to show that the equation can be reduced tofinding the solutions to pythagorean triples,for which there will be infinitely many solutions.The techniques will not be covered in detail in this lesson.
2.Prove that a certain expression is always a perfect square(or perfect cube).
The most straightforward way to do this is to show that it can always be factorid into a perfect square(or cube)!The following result may be uful:If ab is
a perfect square and gcd(a,b)=1,then a and
b must also be perfect squares.
小黄杨Another standard approach is to prove by contradiction.Suppo the expression is not a perfect square,and show that it leads to a contradiction.东莞小吃
If the question is of the type“prove that(some expression)is always a perfect square for all values of n”,then you may want to consider using mathematical induction.
3.Prove that a certain expression is never a perfect square.Simple problems
of this sort can be done by considering the residue of the expression modulo a suitable number,say n.If it is not a quadratic residue of n,then it cannot be a perfect square.
Another common technique involves bounding between concutive squares(or concutive cubes).If n2<x<(n+1)2for an integer n,then x cannot be a perfect square.
Alternatively,if we are able to show that a prime p or an odd power of the prime, p2k+1,divides the expression exactly,then the expression cannot be a perfect square.
4.A problem involving two given expressions that are both perfect squares.If
you are given expression A and expression B and you are told that they are both perfect squares,then you might wish to ask,what conditions must hold for both of them to be perfect squares?If we let A=m2and B=n2,it may be uful to consider A−B=(m+n)(m−n).It may also help to consider if m or n must be odd or even(by taking A and B modulo4,for instance).Considering residue
s of the expressions modulo a suitable number may sometimes also lead you to a solution(or a contradiction).
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If the problem wants you to“find all integers x such that A and B are both perfect squares”,where A and B are expressions involving x,you might want tofind a bound for x.For instance,again we suppo A=m2and B=n2.Then you may want to show that if x exceeds a certain number,then m2<n2<(m+1)2,thus A and B cannot both be perfect squares,a contradiction.
Another approach to this kind of problem is to try to obtain a Pell’s equation from the given expressions.However,that is beyond the scope of our lesson.
5.Equations with unknown exponents If the equation has terms with an unknown
7x,then we may wish to consider residues modulo a suitable number to‘eliminate’the term.For instance,we can u mod7to get rid of7x.Alterna-tively,mod3and mod8are also uful since7x≡1(mod3)and7x≡±1(mod
8).This might lead us to something uful.Alternatively,if there are two or more
terms with unknown exponents,such as3x+4y=5z,then considering mod3and mod4respectively,you
will obtain x and z are even.The equation can then be factorid as a difference of squares.
Residues of Squares,Cubes and Fourth Powers
The following results for perfect squares are easy to verify:
•x2≡0,1(mod4).
More precily,x2≡0(mod4)⇔x is even,and x2≡1(mod4)⇔x is odd.
•x2≡0,1,4(mod8).
In particular,note that x2≡1(mod8)⇔x is odd.
•x2≡0,1,4,9(mod16)
•x2≡0,1(mod3)
•x2≡0,1,4(mod5)
•x2≡0,1,2,4(mod7)
Here are some uful properties of perfect cubes:
经理的职责•x3≡0,±1(mod7)
•x3≡0,±1(mod9)
Finally,for perfect fourth powers:
小学演讲稿•x4≡0,1(mod5)
•x4≡0,1(mod16)
Example1.(Russia MO)Find all pairs of prime numbers(p,q)such that
p3−q5=(p+q)2.
Example2.Let d be any positive integer that is not2,5or13.Prove that at least one of the numbers2d−1,5d−1,13d−1is not a perfect square.
Example3.Find all non-negative integer solutions to the equation3x−y3=1.
Example4.(Putnam1954)Prove that there are no integers x and y such that
项目绩效评价
x2+3xy−2y2=122.
Classroom Problems
1.Let n be a natural number such that2n+1and3n+1are both perfect squares.
Prove that5n+3is composite.
2.If2n+1and3n+1are both perfect squares,prove that n must be divisible by40.
3.Determine all primes p such that5p+12p is a perfect square.
4.Let n be an integer.Prove that if2√
2+2is an integer,then it is a perfect
square.
5.Find all natural numbers n such that28+211+2n is a perfect square.
6.Prove that the system of equations
x2+6y2=z2
6x2+y2=t2
has no non-trivial solutions.
7.Find all integers x and y such that x2+3y and y2+3x are both perfect squares.
8.Find all integers x and y such that x+y,x+2y and2x+y are all perfect squares.
9.Determine all integer solutions to the equation x41+x42+...+x48=20092009.
10.Find all non-negative integers(x,y)satisfying(xy−7)2=x2+y2.
11.Prove that the equation y2=x5−4has no integer solutions.
12.(SMO(S)2004/Round2)Find all pairs of integers(x,y)satisfying the equation
(x2+y2)2=1+16y.
13.Show that the equation2x−1=z m has no integer solutions if x,m>1.
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14.(SMO(O)1998/B3)Do there exist integers x and y such that1919=x3+y4?Justify
your answer.
15.(NTST2007)Find all pairs of nonnegative integers(x,y)satisfying
(14y)x+y x+y=2007.
16.(IMO shortlist2002)What is the smallest positive integer n such that there exists
integers x1,x2,...,x n satisfying x31+x32+...+x3n=20022002?
17.(IMO2006)Determine all pairs(x,y)of integers such that
1+2x+22x+1=y2.
