Orion Crest, Series_1

Chapter 48: Taking out

　　"'Reverse mathematics'?" Surprised, Orion looked at the young Indian, "You proved it using reverse maths? I thought you were studying algebraic number theory."

　　"Algebra is just a tool for studying number theory, it's not the only way ...... I know that might sound a bit unpleasant to you, and this proof method I've come up with will mean that a bunch of mathematicians have gone to the grave together."

　　"...... I'm not upset in any way, can you get to the point quickly?"

　　Diran hung up the board and turned back to Orion and raised an eyebrow smugly.

　　"In a minute!"

　　Just as the young Indian was writing and drawing on the whiteboard, Orion noticed that a number of people were turning their interested eyes this way and approaching towards it.

　　With curiosity, Orion stood next to the poster and followed the Indian guy's thoughts on the proof.

　　His idea was simple.

　　First, assume that the twin prime numbers are limited pairs, and let the largest twin prime pair be (Pn-1, Pn). It is known that the prime numbers up to Pn are finite, and let them be P1, P2.... .Pn-1, Pn.

　　Then construct a large prime number P = (P1P2P3*... *Pn) + 1

　　Clearly,P is not divisible by all prime numbers from P1 to Pn and always has a remainder of 1, so P is prime. Similarly, it can be shown that P-2 = (P1P2P3*... *Pn)-1 is obviously also prime, and division by any prime number from P1 to Pn always differs by 1.

　　Since P is a prime, P-2 is also a prime, and the two form a twin pair of prime numbers.

　　The problem is that P and P-2 form a twin prime pair that is larger than the original "maximum prime pair", thus denying that (Pn, Pn-1) is the maximum twin prime pair.

　　Like climbing a ladder, no matter how big (Pn-1, Pn) is, you can always find a pair of prime numbers bigger than (Pn-1, Pn).

　　Thus, the hypothesis that "twin prime pairs are finite" is disproved, and the opposite is true: "twin prime pairs are infinite".

　　There's a lot more to it than that, but that's the general idea.

　　Orion saw the process he listed on the board from beginning to end. Surprisingly, he didn't refer to any existing research to solve the problem. This kind of thinking outside the box for answers is worth promoting.

　　But ......

　　Orion finally understood why no one was talking to him.

　　"You constructed a large prime P that is indeed guaranteed not to be divisible by a series of primes from P1 to Pn, but only if Pn is the largest prime. Obviously, you've fallen into a logical trap; how do you prove that Pn is the largest known prime?"

　　Diran raised an eyebrow, "Did you not read what I wrote in the first line? In the case of a finite number of twin prime pairs, take the largest twin prime pair (Pn-1, Pn) ......"

　　Orion: "2*3*5*7*11*13+1 = 30031."

　　Upon hearing Orion read out the line of arithmetic, a flash of dawning realisation appeared on the faces of a number of people in the crowd gathered next to him, while some others had obviously guessed it and were already laughing out loud in a small voice at this point.

　　There are also people ......, applauded in small voices.

　　Diran froze, feeling something was wrong, and frowned, "What do you mean?"