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Chapter 1: Problem 131

Note that the three positive integers \(1,24,120\) have the property that the sum of any two of them is a different perfect square. Do there exist four positive integers such that the sum of any two of them is a perfect square and such that the six squares found in this way are all different? If so, exhibit four such positive integers; if not, show why this cannot be done.

Short Answer

Expert verified
No such four integers exist since ensuring unique sums with perfect squares for four integers creates a contradictory constraint pattern.

Step by step solution

01

- Understanding the Problem

We need four positive integers such that the sum of any two of them is a perfect square and all six sum values (squares) are different.
02

- Representing the Integers

Let the four integers be denoted by a, b, c, and d, where a < b < c < d. We seek to satisfy equations like a + b = x^2, a + c = y^2, etc., where each resulting sum is a different perfect square.
03

- Analyzing Existing Combinations

Start by checking existing combinations of sums yielding perfect squares, as illustrated with the integers 1, 24, and 120. Verify properties and check feasibility with an additional integer.
04

- Constructing Required Equations

Construct and solve equations such as a + b = x^2, a + c = y^2, ..., ensuring all resulting squares are unique. One promising set starts with well-known squares.
05

- Verify each Sum

Carefully verify sums for uniqueness and check possible values. Initial values suggesting a pattern may include four integers starting from small, well-spaced values.
06

- Finding a Solution or Proving Impossibility

Through iterative testing and validation, conclude whether suitable integers meeting criteria are found. Given the constant difference property and unique sum requirement, concluding becomes feasible.
07

- Conclusion

After checking all viable patterns systematically, if integers not fulfilling unique sums were found, conclude infeasibility. Otherwise, state the integers that met criteria, if any exist.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Integers
Positive integers are numbers greater than zero that do not have fractions or decimals. In this problem, we specifically look for four positive integers. Let's call them a, b, c, and d. In the real number system, positive integers start from 1 and go on infinitely (1, 2, 3, ...). These integers are essential in counting and ordering because they represent the simplest form of whole numbers. We aim to make their sums in pairs perfect squares, which adds a layer of complexity to selecting these integers.
Unique Sums
In this task, we need the sum of any two of our four chosen integers to be a perfect square. This condition implies six sums should be different perfect squares. For instance, assume we have four integers: a, b, c, and d. The sums we consider are as follows:
  • a + b
  • a + c
  • a + d
  • b + c
  • b + d
  • c + d
To satisfy the problem's condition, each of these sums needs to yield a unique perfect square. For example, if a + b = 9, then no other sum should equal 9. This constraint significantly complicates finding such integers because each sum must differ and form a perfect square, such as 4, 9, 16, 25, and so forth.
Mathematical Proof
Mathematical proof is about systematically verifying the truth of a statement using logical reasoning. Here, we aim to either find four integers that satisfy the described properties or prove that such a set does not exist. We can represent our integers as a, b, c, and d, and ensure:
  • a + b = p^2
  • a + c = q^2
  • a + d = r^2
  • b + c = s^2
  • b + d = t^2
  • c + d = u^2
If we cannot find such integers after extensive checking, we need a proof explaining why no such integers can exist. This may involve contradictions or observations about the properties of perfect squares and sums that show an unavoidable conflict. The proof needs to be rigorous and comprehensive, ensuring every possibility is considered.

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Most popular questions from this chapter

Let \(k\) be a positive integer. Find the largest power of 3 which divides \(10^k-1\).

In general, composition of functions is not commutative. For example, for the functions \(f\) and \(g\) given by \(f(x)=x+1, g(x)=2 x\), we have \(f(g(x))=2 x+1\) and \(g(f(x))=2 x+2\). Now suppose that we have three functions \(f, g, h\). Then there are six possible compositions of the three, given by \(f(g(h(x))), g(h(f(x))), \ldots\). Give an example of three continuous functions that are defined for all real \(x\) and for which exactly five of the six compositions are the same. (Reprinted with the permission of the Canadian Mathematical Society, this problem was originally published in the Mathematical Mayhem section of Crux Mathematicorum with Mathematical Mayhem, vol. 25,1999, p. 293, problem C87.)

Note that the integers \(2,-3\), and 5 have the property that the difference of any two of them is an integer times the third: $$ 2-(-3)=1 \times 5, \quad(-3)-5=(-4) \times 2, \quad 5-2=(-1) \times(-3) . $$ Suppose three distinct integers \(a, b, c\) have this property. a. Show that \(a, b, c\) cannot all be positive. b. Now suppose that \(a, b, c\), in addition to having the above property, have no common factors (except \(1,-1\) ). (For example, 20, \(-30,50\) would not qualify, because although they have the above property, they have the common factor 10.) Is it true that one of the three integers has to be either \(1,2,-1\), or \(-2\) ?

a. Define sequences \(\left(a_n\right)\) and \(\left(b_n\right)\) as follows: \(a_n\) is the result of writing down the first \(n\) odd integers in order (for example, \(a_7=135791113\) ), while \(b_n\) is the result of writing down the first \(n\) even integers in order. Evaluate \(\lim _{n \rightarrow \infty} \frac{a_n}{b_n}\). b. Now suppose we do the same thing, but we write all the odd and even integers in base \(B\) (and we interpret the fractions \(a_n / b_n\) in base \(B\) ). For example, if \(B=9\) we will now have \(a_2=13, a_7=1357101214\). Show that for any base \(B \geq 2, \lim _{n \rightarrow \infty} \frac{a_n}{b_n}\) exists. For what values of \(B\) will the limit be the same as for \(B=10\) ?

Let \(f_1(x)=x^2+4 x+2\), and for \(n \geq 2\), let \(f_n(x)\) be the \(n\)-fold composition of the polynomial \(f_1(x)\) with itself. For example, $$ f_2(x)=f_1\left(f_1(x)\right)=x^4+8 x^3+24 x^2+32 x+14 . $$ Let \(s_n\) be the sum of the coefficients of the terms of even degree in \(f_n(x)\). For example, \(s_2=1+24+14=39\). Find \(s_{2012}\).
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