And perhaps even find your hidden prime factors

Neil L. is a Leprechaun. He has visited me every St. Patrick’s Day since I began the blog in 2009. In fact he visited me every St. Patrick’s Day before then, but I never noted him. Sometimes he comes after midnight the night before, or falls asleep on my sofa waiting for me to rise. But this time there was no sign of him as I came back from a long day of teaching and meetings and went out again for errands.

Today Ken and I wish you all a Happy St. Patrick’s Day, and I am glad to report that Neil did find me.

When I came back I was sorting papers and didn’t see him. I didn’t know he was there until I heard,

Top o’ the evening to ye.

Neil continued as he puffed out some green smoke: “I had some trouble finding you this year. Finally got where you were—good friends at your mobile provider helped me out.” I was surprised, and told him he must be kidding. He answered, “Of course I always can find you, just having some fun wi’ ye.” Yes I agreed and added that I was staying elsewhere. He puffed again and said “yes I understand.”

I said I had a challenge for him, a tough challenge, and asked if he was up for it. He said, “Hmmm, I do not owe you any wishes, but a challenge… Yes I will accept a challenge from ye, any challenge that ye can dream up.” He laughed, and added, “we leprechauns have not lost a challenge to a man for centuries. I did have a cousin once who messed up.”

## The Cousin’s Story

I asked if he would share his cousin’s story, and he nodded yes. “‘Tis a sad story. My cousin was made a fool of once, a terrible black mark on our family. Why, we were restricted from any St Patrick Day fun for a hundred years. Too long a punishment in our opinion—the usual is only a few decades. Do ye want to know what my cousin did? Or just move on to the challenge? My time is valuable.”

I nodded sympathetically, so he carried on.

“One fine October day in Dublin me cousin was sitting under a bridge—under the lower arch where a canalside path went.

“He spied a gent walking with his wife along the path but lost in thought and completely ignoring her. He thought the chap would be a great mark for a trick but forgot the woman. She spied him and locked on him with laser eyes and of course he was caught—he could not run unless she looked away.

“He tried to ply her with a gold coin but she knew her leprechaun lore and was ruthless. He resigned himself to granting wishes but she would not have that either. With her stare still fixed she took off her right glove, plucked a shamrock, and lay both at his feet for a challenge. A woman had never thrown a challenge before, and there was not in the lore a provision for return-challenging a woman. So my cousin had to accept her challenge. It came with intense eyes:

“I challenge you to tell the answer to what is vexing and estranging my husband.”

“Aye,” Neil sighed, “you or I or any lad in the face of such female determination would be reduced to gibberish, and that is what me cousin blurted out:

${i^2 = j^2 = k^2 = ijk = -1.}$

“The gent looked up like the scales had fallen from his eyes, and he embraced his wife. This broke the stare, and my cousin vanished in great relief. And did the gent show his gratitude? Nay—he even carved that line on the bridge but gave no credit to my cousin.”

I clucked in sympathy, and Neil seemed to like that. He put down his pipe and gave me a look that seemed to return comradeship. Then I understood who the “cousin” was. Not waiting to register my understanding, he invited my challenge as a peer.

## My Challenge

I had in fact prepared my challenge last night—it was programmed by a student in my graduate advanced course using a big-integer package. Burned onto a DVD was a Blum integer of one trillion bits. I pulled it out of its sleeve and challenged Neil to factor it. The shiny side flashed a rainbow, and I joked there could really be a pot of gold at the end of it.

Neil took one puff and pushed the DVD—I couldn’t tell how—into my MacBook Air. The screen flashed green and before I could say “Jack Robinson” my FileZilla window opened. Neil blew mirthful puffs as the progress bar crawled across. A few minutes later came e-mail back from my student, “Yes.”

I exclaimed, “Ha—you did it—but the point isn’t that you did it. The point is, it’s doable. You proved that factoring is easy. Could be quantum or classical but whatever—it’s practical.”

Neil puffed and laughed as he handed me back the suddenly-reappeared disk and said, “Aye, do ye really think I would let your lot fool me twice?”

I replied, “Fool what? You did it—that proves it.”

“Nay,” he said, “indeed I did it—I cannot lie—but ye can’t know how I did it enough to tell whether a non-leprechaun can do it. And a computer that ye build—be it quantum or classical or whatever—is a non-leprechaun.”

It hit me that a quantum computer that cannot be built is a leprechaun, and perhaps Peter Shor’s factoring algorithm only runs on those. But I wasn’t going to be distracted away from my victory.

“How can it matter whether a leprechaun does it?” Neil retorted that he didn’t have to answer a further challenge, “it’s not like having three wishes, you know.” But he continued, “since ye are a friend, I will tell ye three ways it could be, and you can choose one ye like but know ye: it could still be a fourth way.

1. “I could have been around when your student made the number, even gone back in time to see it. I did take a long time to find ye, did I not?
2. “Since y’are not a woman I have a return challenge, and I don’t have to give it after yours or even tell ye. I get some control and can influence you to give instructions that will lead to a particular number I am prepared for. We leprechauns do that with choices of RSA keys all the time.
3. “Everything in your world that you create by rules is succinct. Of course so was that number. Factoring of succinct numbers is easy—indeed in this world, everything is easy.

“And I left ye a factor, but your student already had it, so I left ye no net knowledge at all.” And with a puff of smoke, he was gone.

## Open Problems

Did I learn anything from the one-time factoring of my number? Happy St. Patrick’s Day anyway.

[moved part of dialogue at end from 2. to 1.]