Think of numbers as LEGO bricks. We can build any number by adding up smaller numbers. But what if we only allowed ourselves to use very specific LEGO bricks – ones shaped like prime powers (numbers like 4, 8, 9, 16, 25, and so on, created by raising a prime number to a power of 2 or higher)? Would we still be able to build every number? And if so, how many of these special bricks would we need?
That’s the puzzle that Julius Stricker at an undisclosed institution (the paper does not state the institution) has been wrestling with, and the answer is surprisingly elegant: five. According to a new conjecture, every number greater than 23 can be expressed as the sum of no more than five prime powers. In other words, any number you pick after 23 can be built using a maximum of five of these prime power ‘bricks’.
The Prime Power Toolkit
Before diving deeper, let’s clarify what a ‘prime power’ actually is. It’s simply a prime number (like 2, 3, 5, 7, 11…) raised to an integer power of 2 or greater. So, 22 = 4, 33 = 27, and 52 = 25 are all prime powers. What’s important is that we specifically exclude prime numbers themselves (e.g., 2, 3, 5, 7) from being considered prime powers in this context. It’s all about those exponents!
Why is this restriction important? It focuses our attention on a subset of numbers that are, in a way, even more ‘structured’ than prime numbers themselves. They represent the multiplicative building blocks of numbers raised to a higher power, forcing us to think about how numbers can be constructed not just from simple addition but through a combination of multiplication (the prime base raised to a power) and addition.
Computational Brute Force (and Elegance)
Stricker’s conjecture isn’t just some abstract idea scribbled on a napkin. It’s backed by serious computational muscle. He exhaustively checked all numbers up to 10 million, meticulously searching for combinations of prime powers that would add up to each number. The result? Not a single number required more than five prime power summands.
But 10 million, while a large number, is just a tiny sliver of the infinite number line. So, to bolster the conjecture further, Stricker sampled 100,000 additional numbers, starting from 1.5 million and stretching all the way to 10 billion. Again, the conjecture held firm: five prime powers were always enough.
To put this in perspective, imagine trying to crack a complex safe. Exhaustively checking every combination is often impossible. Instead, you might try a few targeted approaches based on what you know about the lock’s mechanism. Stricker’s computational work is a bit of both: he exhaustively checked a large initial range, then strategically probed much larger numbers to see if the pattern held. This combined approach gives us a high degree of confidence in the conjecture, even though a formal mathematical proof remains elusive.
Why Five? The Counterintuitive Nature of Prime Powers
What makes this conjecture so interesting is its counterintuitive nature. We know from other areas of number theory that prime numbers themselves – which are more common than prime powers – can be used to build numbers. Goldbach’s Conjecture, for example, (still unproven!) states that every even number greater than 2 can be expressed as the sum of two primes. The Weak Goldbach Conjecture (now a theorem) says that every odd number greater than 5 can be expressed as the sum of three primes.
So, if we need up to three *prime numbers* (a relatively dense set) to build most numbers, why do we only need up to five *prime powers* (a much sparser set)? Shouldn’t it take *more* of the rarer building blocks?
The key, Stricker argues, lies in what he calls the “combinatorial creativity” of prime powers. Prime numbers are, in a sense, rigid. They’re always just ‘p’ (p1). Prime powers, on the other hand, have exponents. This gives them a flexibility that prime numbers lack. Think of it like this: prime numbers are like standard-sized LEGO bricks, while prime powers are like a set of bricks with varying heights and widths. This variety allows you to fill gaps more efficiently.
Imagine you are tiling a floor. If you only have one size of tile, you might have to cut some tiles to fit the edges, leading to waste. But if you have a variety of tile sizes, you can likely cover the floor much more efficiently with fewer cuts. Prime powers, with their varying exponents, offer a similar advantage in the world of numbers.
The Race to Infinity: Primes vs. Prime Powers
Stricker uses a vivid analogy to describe this phenomenon: “It is as if the race starts in ‘rear gear’ compared to primes, but then accelerates to ‘light speed’ towards infinity due to the sheer volume of available sum configurations.”
Initially, prime powers might seem to be at a disadvantage because they are less frequent. But as numbers grow larger, the sheer number of possible combinations of prime powers explodes, allowing them to catch up and even surpass the efficiency of prime numbers in representing other numbers.
Waring-Goldbach and the Power of Mixing
This conjecture also touches on the famous Waring-Goldbach Problem, which explores how numbers can be represented as sums of powers of primes. However, the Waring-Goldbach Problem typically focuses on sums of primes raised to a *fixed* power (e.g., sums of squares of primes, or sums of cubes of primes).
Stricker’s conjecture is unique because it allows for *mixed* prime powers. You can use squares, cubes, fourth powers, fifth powers, and so on, all in the same sum. This added flexibility makes the problem significantly more complex to analyze mathematically, but it also seems to be the key to its surprising efficiency.
It’s like having a toolbox filled with every size of wrench imaginable. While figuring out which wrench to use for each bolt might be more complicated, you’re much more likely to find the perfect fit quickly. The Waring-Goldbach approach, in contrast, is like having a limited set of wrench sizes. You might eventually get the job done, but it could take longer and require more effort.
The Road Ahead: Proving the Unprovable?
While the computational evidence for Stricker’s conjecture is compelling, a formal mathematical proof remains the holy grail. Such a proof would likely require sophisticated tools from analytic number theory, such as the Hardy-Littlewood circle method. These methods aim to demonstrate that the number of ways to represent a number as a sum of prime powers is non-zero as the number approaches infinity.
The challenge is that the gaps between prime powers can be enormous, reaching trillions or even quintillions in higher ranges. This makes it computationally impossible to exhaustively check every number. Instead, a proof would need to rely on theoretical arguments to bridge the gap between the computational evidence for smaller numbers and the behavior of numbers at infinity.
Stricker’s work provides a tantalizing glimpse into the hidden structure of numbers. It suggests that even seemingly sparse sets of numbers, like prime powers, can possess surprising combinatorial power. Whether he or another mathematician can ultimately prove this conjecture remains to be seen, but the journey promises to be a fascinating one.
