Question

An arithmetic progression of integers \(a_{n}\) is one in which \(a_{n}=a_{0}+n d\), where \(a_{0}\) and \(d\) are integers and \(n\) takes successive values \(0,1,2, \ldots .\) (a) Show that if any one term of the progression is the cube of an integer then so are infinitely many others. (b) Show that no cube of an integer can be expressed as \(7 n+5\) for some positive integer \(n\)

Short answer

(a) If one term in an arithmetic progression is a cube, infinitely many others will be cubes too. (b) No cube of an integer can be expressed as \(7n + 5\).

Step-by-step solution

Understand Arithmetic Progression

Identify the general form of an arithmetic progression: \(a_{n} = a_{0} + nd\). Here, \(a_{0}\) is the first term and \(d\) is the common difference.

Given Condition of a Cube

Assume \(a_{k} = k\) is a cube of an integer, where \(a_{k} = a_{0} + kd\). Let this cube be \(m^3\) for some integer \(m\). Therefore, \(a_{0} + kd = m^3\).

Rearrange the Equation

Isolate \(a_{0}\): \(a_{0} = m^3 - kd\).

Formulate Other Cubes in Progression

Substitute a different value of \(n\) in \(a_n = a_0 + nd\) such that another term also becomes the cube of an integer. Let \(a_{j} = n^3\) for some integer \(n\). Then, \(a_{0} + jd = n^3\).

Simplify the Second Equation

Rewrite \(a_{0} + jd = n^3\) as \(m^3 - kd + jd = n^3\). Simplify to find \(jd - kd = n^3 - m^3\), giving us \(d(j - k) = (n - m)(n^2 + nm + m^2)\)

Conclude Infinitely Many Cubes

Since \(m\) and \(n\) are arbitrary integers, infinitely many integers can satisfy the given conditions, proving that infinitely many terms in the progression are cubes of integers.

Verify No Cube Form 7n + 5

Assume a cube \(x^3\) can be expressed as \(7n+5\). Consider the last digit of cubes modulo 7. Possible residues for \(x^3\) modulo 7 are \(0, 1, 6\).

Check Possible Residues

List the cubes of integers modulo 7: \(0^3 = 0, 1^3 = 1, 2^3 = 8 \equiv 1, 3^3 = 27 \equiv 6, 4^3 = 64 \equiv 1, 5^3 = 125 \equiv 6, 6^3 = 216 \equiv 6\).

Residue Comparison

Notice no residue of form \(0,1 \) or \(6\) is equivalent to 5 modulo 7, therefore no integer cube can be expressed as \(7n + 5\).

Key concepts

cube of an integer

A cube of an integer, often referred to as a perfect cube, is a number that can be expressed as the product of an integer multiplied by itself twice. Mathematically, if we denote the integer as 'm', then its cube is represented as \(m^3 = m \times m \times m\). For example, the cube of 2 is 8, since \(2^3 = 2 \times 2 \times 2 = 8\). Perfect cubes are important in various branches of mathematics, especially in number theory and algebra. They exhibit interesting properties, such as being able to be factorized into three equal factors. When working with sequences, such as arithmetic progressions, cubes of integers can lead to fascinating patterns and behaviors that we explore further.

modulo arithmetic

Modulo arithmetic, also known as 'clock arithmetic', is a system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value. The general setup uses the concept of a modulus. For instance, consider a modulus of 7. In modulo 7 arithmetic, any integer is replaced by its remainder when divided by 7. For example, 15 modulo 7 is 1 because 15 divided by 7 leaves a remainder of 1. This concept is crucial in proofs and problems dealing with periodicity and cyclic behaviors.

When applying this to cubes of integers, one assesses the remainder when these cubes are divided by 7 (or any other modulus). For example, possible residues for \(m^3 \) modulo 7 are 0, 1, and 6. These specific residues arise because when evaluating the cubes of numbers (like 0 through 6) modulo 7, only 0, 1, and 6 appear as remainders.

Understanding modulo arithmetic can provide insights into patterns and can reveal properties, such as why no cube of any integer can be expressed in a specific form like \(7n + 5\).

integer sequences

Integer sequences are ordered lists of integers that follow specific rules or patterns. A common example of an integer sequence is an arithmetic progression (AP). In an AP, each term after the first is obtained by adding a fixed number, known as the common difference (\(d\)), to the previous term. This can be represented as \(a_n = a_0 + nd\), where \(a_0\) is the first term and \(n\) is a non-negative integer.

For example, the sequence 2, 5, 8, 11,... is an arithmetic progression where \(a_0 = 2\) and \(d = 3\).Integer sequences are pivotal in various fields of mathematics and provide fundamental structures in number theory and combinatorics.

In relation to the problem, we can see that if one term in an arithmetic progression of integers is a cube of an integer, then by manipulating the properties of the common difference and integer cubes, we can find infinitely many other terms in the sequence that are also perfect cubes. This leads to deep insights into the nature of integer sequences and their inherent patterns.