Orion Crest, Series_2

Chapter 17: Presumably on purpose

　　Hearing the compliment, the corners of Vera's mouth quirked up with a touch of happiness. Standing next to Orion, she said softly.

　　"Your guess is correct, Collatz Conjecture is a number theory problem as well as a complex analysis problem ......"

　　Back in 1994, L. Berg and G. Meinardus proved that the 3n+1 conjecture is equivalent to the functional equation h(z cubed) = h(z^6) + {h(z squared) + λh(λz squared) + λsquared h(λsquaredz squared)}/3z (where λ = e^(2πi/3)) in the unit disc {z:|z|<1} with an analytic functional solution presented as h(z ) = h0 + h1z/(1-z) form. (h0 and h2 are complex constants.)

　　And on top of that, D. Schleicher et al. proved in 1998 that any integer function h(z) such that g(z) = z/2 + (1 - cosπz)(z + 1/2)/2 + 1/π(1/2 - cosπz) sinπz + h(z) sin square πz satisfies: NCΦ(g).

　　Based on these two conclusions, Vera proved the existence of an entire function h(z) by constructing an ingenious transcendental entire function such that for every branch D of g(z), Φ(g) in the above conclusions that contains some positive integer, there exists z0 ∈ D such that {g^ok(z0)}∞/k=1 converges to 1. From this it is easy to introduce that the Collatz Conjecture holds!

　　"Excellent proof ......" said Orion from the bottom of his heart with a pleased smile on his face, "excellent to my surprise."

　　From the summer of '16, it was now the end of '17. He was pleased to see that his students had grown. Moreover, he was happy to see that the theory of "group construction method" that he had constructed for the problem of additive number theory did not stop at Goldbach's Conjecture, and also inherited by his students.

　　He now has some experience of what kind of feeling the system depicts as "joy".

　　"...... Thanks to your guidance." Curving the corners of her mouth, Vera spoke humbly, her eyes filled with gratitude.

　　Although the process was completed by her, the entire proof idea was provided by Orion. From the milestones of her presentation at Berkeley at the beginning of the year, to the current collaboration with Franklin, Hardy and the two of them to complete the final proof, all the work was centred around this line of thought.

　　Orion smiled, "There's no need to be modest, all I'm providing is the direction, it's you who runs to the finish line."

　　"...... About this paper, I suggest you to submit it in Annals of Mathematics, but the editorial board is on holiday these days, you can publish the paper on Arxiv first ......"

　　Orion certainly has a bit of his own selfishness in mind for making this suggestion. According to the inference he had made when solving Goldbach's Conjecture, the system's principle for task completion judgement was based on the time the paper was made public. So as long as the paper was listed on Arxiv, it could be considered task completion.

　　After hearing Orion's words, Vera nodded seriously.

　　"I know, I'll do it right away."

　　Orion smiled, "That's it then, as for the arithmetic on the blackboard, I'll just erase it for you. ...... Thank you for your gifts."

......

　　After a full day of fun at the Institute for Advanced Study in Princeton, Orion returned home with the small gifts that his students and colleagues had given him.

　　Sitting down on the couch next to the fireplace, closing his eyes and silently chanting a system, Orion sank his consciousness into the system space. Walking over to the translucent holographic panel, two lines of transparent text floated in front of him.