An Invitation to Math Olympiad 1 – Plato’s Dianoa and the identity of Sophie Germain

Once upon a time there was a beautiful problem.

Is there a number $n$ such that $n^4 + 4^n$ is a prime number?

Clearly $ n = 1 $ works. After all, $1^4 + 4^1$ equals $5$. What about $n = 2$? That does not work. $2^4 + 4^2$ is the composite number 32.

Prime numbers are positive integers with exactly $2$ distinct divisors (not more and not less). This definition immediately rules out $1$ as it has exactly $1$ divisor: $1$ itself. Other examples of prime numbers are $3$, $5$, $7$ etc.

Can you find all the prime numbers from $1$ to $100$? What about $1$ to $1000$? Is there a pattern in which the number of primes is increasing?

Composite numbers, on the other hand, are integers with more than $2$ divisors. For example $10$ is a composite number as it has $4$ distinct divisors: $1, 2, 5, 10$.

Composite numbers can be factorized. That is, they can be written as a product of two or more prime numbers. There is a technical theorem known Euclid’s unique factorizarion theorem, which says that any composite number can be uniquely factorized into its prime factors (you can rearrange the factors). For example:

$$ 10 = 2 \times 5 $$

It is difficult to tell, if a number is prime or composite. For example can you quickly figure out if 2024202320222021 is a prime number or a composite number?

One way to ‘prove’ that a number is composite is to write it as a product two numbers (not necessarily prime numbers) such that none of these factors are $1$. It is usually hard to do it. But, in the case of our problem, this is doable.

Lets revisit the expression.

$$ n^4 + 4^n $$

Clearly if $n$ is even, this number is divisble by $2$. Apart from $2$ itself, all other prime numbers are odd. It is also easy to see that $ n^4 + 4^n $ is greater than $2$. Hence it is not a prime number (as it can be written as $2$ times some number).

The tricky case happens when $n$ is odd. In order to analyze this situation, we rewrite $n$ as $2k + 1$. This is something that we often do to solve number theory problems. Basically every odd number can be thought of as an even number + 1. Any even number is twice of ‘some-number’. Now denote this ‘some-number’ as ‘k’. Hence the even number can be written as $2 \times k $ and the odd number as $2k + 1$ (as it is one more than the even number).

Replacing $n$ by $2k + 1$, we immediately gain further insight into the problem.

$$ (2k + 1)^4 + 4^{2k + 1}$$.

This expression is actually factorizable! That is, you can write it as a product of two numbers, where if $k \neq 0$, none of those two factors is $1$. This factorizarion is called the Sophie Germain identity.

$ a^4 + 4b^4 = (a^2 + 2b^2)^2 – 2 \times a^2 \times 2b^2 $

But the later expression can be re-written as

$$ (a^2 + 2b^2)^2 – (2ab)^2 $$

This yields a factorization using ‘difference of two squares identity’:

$$ (a^2 + 2ab + 2b^2) \times (a^2 – 2ab + 2b^2) $$

All that remains to be done is to write $ (2k + 1)^4 + 4^(2k + 1) $ as $ a^4 + 4b^4 $. This works out because $ (2k + 1)^4 + 4^{2k + 1} $ = $ (2k + 1)^4 + 4 \cdot 4^{2k} $. This equals to $ (2k + 1)^4 + 4 \cdot 2^{4k} $ = $ (2k + 1)^4 + 4 \cdot (2^k)^4 $. Therefore we can identify $ (2k + 1) = a, 2^k = b $ and conclude that $ (2k + 1)^4 + 4^(2k + 1) $ is of the form $ a^4 + 4b^4 $.

One final remark for this section is related to Plato’s Divided Line. Plato was one of the most celebrated Greek philosophers from antiquity. In his seminal work ‘Republic’, Plato presents the idea of the Divided Line as a dialog between Glaucon and Socrates.

The ‘Divided Line’ essentially describes Plato’s opinion about the affections of human psyche. First two consists of shadows and reflections of physical things and the physical things themselves. Then comes the third level of Dianoia, where according to Plato, the intelligible world begins. In particular, in this level, one recognizes the ‘form’ in the ‘real objects’.

Consider the thought-exercise that we explained above. We looked at $ n^4 + 4^n $ and recognized that it is of ‘form’ $ a^4 + 4b^4 $ in the special case where $ n = $ odd. Much of mathematics is about playing this game of recognizing the general ‘form’ of something specific. In a sense this exercise leads us a higher level of human-ness compared to ‘things’ and ‘opinions’.


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