In the Harry Potter universe it is established that the number seven has magical properties, and is the most powerfully magical number. But what does that mean? What are these 'magical properties' and how do they manifest?
To clarify, I'm not interested in the reasons why seven was chosen as the most magically powerful, but rather what it means for a number to be magically powerful, because beyond cropping up a bunch of times I'm unsure what the innate magical property of the number seven is.
Numerology is a real-life belief that certain numbers hold more or less magical, divine, or otherwise supernatural power. In Harry Potter it appears to be an established fact, and study of this phenomenon is called Arithmancy and is taught at Hogwarts.
The description of the discovery of the "theorem" of the power of seven from the PS3 game Wonderbook: Book of Spells suggests some sort of hard scientific process is involved.
It was one day, during breakfast, that Wenlock wrote the immensely famous theorem that supported the magical properties of the number seven. She wrote it down on what she thought was the back of an envelope, in the usual invisible ink. That same day, she posted a letter to her cousin, using what she later realised was the envelope in which she had written the theorem.
So why, mathematically, scientifically, is seven more powerful than other numbers (and why is six so afraid of it)? Oddly enough, the precise theoretical underpinnings of the mystical forces driving the universe were left out of the children's video game.
I'm being flip, but the answer for now is... who knows? You may as well ask why thinking happy thoughts and saying Latin words make white animals shoot out of a piece of wood with a bird's feather in it. Magic in the Harry Potter is magical, defying hard explanations. Other than the fact that we know there is such a thing as "magical theory," the books go into little detail about how magic actually works (for the best, in my opinion).
Throughout history, many numbers have been considered magical in and of themselves, and the number seven is one of them.
From the Bible, the world was created in 7 days, anyone who dares to kill Cain will suffer vengeance seven times over, Pharaoh dreamed of 7 fat cows followed by and devoured by 7 thin cows (7 bountiful years followed by 7 bad years), and Jacob worked 7 years for Leah and 7 years for Rachel. I'm sure there are more.
In more modern folk belief, the 7th son is supposed to be something special, and the 7th son of a 7th son is miraculous (which is the idea in Orson Scott Card's The Tales of Alvin Maker series).
Other examples are the 7 cardinal virtues and 7 deadly sins, the 7 hells and 7 heavens of Islam, and the Seven Sleepers, and the Seven Sisters (the Pleiades).
It seems likely that J.K. Rowling drew on this tradition in Harry Potter- probably especially the 7th son idea. I don't, however, recall any specific effects of the number 7 mention in Harry Potter - unlike in Terry Pratchett's Discworld books, where the number 8 has numerous powerful effects.
Within Harry Potter itself I can only think of a single other instance aside from Voldemort's Horcruxes where the number 7 demonstrated a magical effect. In The Half Blood Prince, Harry is using an old potions book with notes in, and by doing so is gaining a reputation as a Potions genius. While making a Draught of Living Death he comes across these note;
According to the book, he had to stir counter-clockwise until the potion turned clear as water. According to the addition the previous owner had made, however, he ought to add a clockwise stir after every seventh counter-clockwise stir. Could the old owner be right twice?
So in this instance the number seven is manifesting as improved potion making techniques. Harry goes on to win the contest Slughorn is staging at this point in the book.
