3.3 Early number theory and figurate numbers

Number theory studies the properties of integers, with a focus on primes, divisibility, and patterns. The Pythagoreans were among the earliest to treat numbers themselves as objects worth studying, and their work on figurate numbers and special number sets laid groundwork that Euclid later formalized in the Elements.

Figurate numbers are integers that can be arranged into regular geometric shapes using evenly spaced dots. The Pythagoreans explored triangular, square, and pentagonal numbers in depth, discovering surprising relationships between them. These discoveries fed directly into broader number theory concepts like perfect and amicable numbers.

Number Theory Foundations

Fundamental Concepts of Number Theory

Number theory asks: what patterns and properties emerge when you study whole numbers? The ancient Greeks, especially the Pythagoreans, were fascinated by this question. They didn't just use numbers for counting or measurement; they believed numbers had intrinsic qualities worth investigating.

Two key types of sequences show up repeatedly in early number theory:

Arithmetic progressions are sequences where each term increases by a constant difference $d$:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference. For example, the sequence 3, 7, 11, 15 has $d = 4$.

Geometric progressions are sequences where each term is multiplied by a constant ratio $r$:

$a_n = a_1 \cdot r^{n-1}$

For example, the sequence 2, 6, 18, 54 has $r = 3$.

These two sequence types matter because figurate numbers, as you'll see below, grow according to patterns that can be described using arithmetic progressions.

Applications and Historical Significance

The Babylonians worked with number patterns in their astronomical calculations, and the Greeks built on this tradition. Euclid's Elements (c. 300 BCE) devoted Books VII through IX to number theory, including proofs about primes and perfect numbers that still hold up today.

Some useful formulas the Greeks developed for these sequences:

• Sum of an arithmetic sequence: $S_n = \frac{n}{2}(a_1 + a_n)$
• Sum of a geometric sequence: $S_n = \frac{a_1(1 - r^n)}{1 - r}$ for $r \neq 1$

The arithmetic sum formula is traditionally attributed to a method the young Gauss used centuries later, but the underlying idea traces back to ancient Greek mathematics. Geometric sums appear in Euclid's work on proportions and later in problems involving compound growth.

Figurate Numbers

Types and Properties of Figurate Numbers

Figurate numbers are integers that can be represented as regular arrangements of dots. The Pythagoreans studied them by literally laying out pebbles (psephoi) on the ground and observing the patterns.

Triangular numbers form equilateral triangles. Start with 1 dot, then add a row of 2 beneath it, then a row of 3, and so on. The nth triangular number is:

$T_n = \frac{n(n+1)}{2}$

• First few: 1, 3, 6, 10, 15, 21

Square numbers form perfect squares. The nth square number is simply:

$S_n = n^2$

• First few: 1, 4, 9, 16, 25, 36

Pentagonal numbers form nested pentagons. The formula is less intuitive:

$P_n = \frac{n(3n-1)}{2}$

• First few: 1, 5, 12, 22, 35, 51

Notice that the differences between consecutive triangular numbers are 2, 3, 4, 5... (increasing by 1 each time). The differences between consecutive square numbers are 3, 5, 7, 9... (the odd numbers). The differences between consecutive pentagonal numbers are 4, 7, 10, 13... (an arithmetic progression with $d = 3$). This pattern generalizes: for any type of figurate number, the differences between consecutive terms form an arithmetic progression.

Historical Context and Mathematical Relationships

The Pythagoreans used a concept called the gnomon to understand how figurate numbers grow. A gnomon is the shape you add to one figurate number to get the next one of the same type.

For square numbers, this is especially visual. To go from a 2×2 square (4 dots) to a 3×3 square (9 dots), you add an L-shaped gnomon of 5 dots along two edges. Each successive gnomon for square numbers contains the next odd number of dots (3, 5, 7, 9...), which is why the sum of the first $n$ odd numbers always equals $n^2$. That's a genuinely elegant result.

The Pythagoreans also discovered connections between figurate number types:

• Every square number equals the sum of two consecutive triangular numbers. For example, $S_4 = 16 = 6 + 10 = T_3 + T_4$.
• The difference between consecutive pentagonal numbers forms an arithmetic sequence with common difference 3.

These relationships aren't coincidences. They follow directly from the formulas. You can verify the square-triangular relationship algebraically: $T_{n-1} + T_n = \frac{(n-1)n}{2} + \frac{n(n+1)}{2} = \frac{n^2 - n + n^2 + n}{2} = n^2 = S_n$.

Special Number Sets

Perfect and Amicable Numbers

A perfect number equals the sum of its proper divisors (all divisors except the number itself).

• 6 is perfect: $1 + 2 + 3 = 6$
• 28 is perfect: $1 + 2 + 4 + 7 + 14 = 28$
• The next two are 496 and 8128

Euclid proved in Elements Book IX, Proposition 36 that whenever $2^n - 1$ is prime (these primes are now called Mersenne primes), the number $2^{n-1}(2^n - 1)$ is perfect. You can check: when $n = 2$, $2^2 - 1 = 3$ is prime, so $2^1 \times 3 = 6$ is perfect. When $n = 3$, $2^3 - 1 = 7$ is prime, so $4 \times 7 = 28$ is perfect.

Much later, Euler proved the converse: every even perfect number has this form. Whether any odd perfect numbers exist remains an open question to this day. None have been found, and there are strong constraints on what they'd have to look like, but no one has proven they're impossible.

Amicable numbers come in pairs where each number equals the sum of the other's proper divisors. The smallest pair is 220 and 284:

• Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110. Their sum is 284.
• Proper divisors of 284: 1, 2, 4, 71, 142. Their sum is 220.

The Pythagoreans knew this pair and considered it a symbol of friendship.

Historical Significance and Modern Applications

The Pythagoreans attached deep symbolic meaning to these numbers. They considered 6 a number of creation (the world was supposedly made in 6 days), and amicable pairs represented ideal friendship or harmony.

Beyond the Greeks, Islamic mathematicians made notable advances. Thābit ibn Qurra (9th century CE) developed a rule for generating amicable pairs, which was the first systematic method for finding them beyond the classical pair of 220 and 284.

In the modern era, perfect numbers connect to one of the biggest open problems in mathematics: the search for Mersenne primes. Each new Mersenne prime yields a new (enormous) even perfect number. And the study of divisor sums that underlies both perfect and amicable numbers has influenced areas of modern number theory, including the design of certain cryptographic algorithms and pseudorandom number generators.