Multiply two large prime numbers together, and the calculation is fast — any computer manages it instantly, no matter how large the primes are. Take that result and try to work backward, figuring out which two primes were multiplied to produce it, and there's no known fast way to do it at all. The best known methods get dramatically slower as the numbers get larger, to the point of being practically impossible for large enough primes even with enormous computing power. That one-directional difficulty is the entire mathematical foundation RSA encryption, one of the most widely used methods for securing information online, is built on.
Easy one way, brutally hard the other
This asymmetry — easy to compute forward and effectively impossible to reverse without extra information — is called a trapdoor function, and prime factorisation is the clearest, most famous example. RSA encryption, developed in 1977 and named for its inventors Rivest, Shamir, and Adleman, generates a public key built from the product of two large, secretly chosen primes, and a private key built from those primes themselves. Anyone can use the public key to encrypt a message, but reversing that encryption without the private key requires factoring the enormous number back into its two original primes — the exact hard direction of the same asymmetry.
Why the size of the primes is the entire security budget
The security of RSA encryption comes down almost entirely to choosing primes large enough that even the fastest known factoring algorithms, run on the most powerful available computers, would take an impractically long time — often estimated in years, decades, or far longer — to factor the resulting number back into its two primes. As computing power increases over time, the minimum "safe" size of the primes used has had to grow correspondingly, which is why encryption standards periodically get updated to require longer keys.
What we're still unsure about
It has never been mathematically proven that factoring large numbers back into their prime components is inherently hard in the strict computational sense — only that no fast classical method has been discovered despite decades of dedicated effort, which is a different, weaker claim. Compounding that uncertainty, quantum computers running Shor's algorithm, first described in 1994, could in principle factor large numbers efficiently if a sufficiently large, stable quantum computer is ever built — a real enough prospect that cryptographers are already developing "post-quantum" encryption methods not based on the factoring problem, well before such a machine exists, precisely because the security RSA relies on has never been guaranteed, only untested at scale.
This sits inside Prime Numbers & Divisibility, one of seven topics in Arithmetic, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.