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“Jesus.”

“Unfortunately it’s less straightforward with Ingram. I’m convinced he’s the brains behind the whole thing. Otherwise all of this doesn’t add up. But I don’t have a smoking gun, not yet, which makes the whole operation risky. There’s always a chance — though I wouldn’t bet on it — that the file the hacker downloaded has something specific on him. But it’s impossible to crack — a goddamn R.S.A. encryption.”

“So what are you going to do?”

“Tighten the net. Show the world that our very own co-workers are in cahoots with criminal organizations.”

“Like the Spiders.”

“Like the Spiders. And plenty of other bad guys. It wouldn’t surprise me if they were involved in the killing of your professor in Stockholm. They had a clear interest in seeing him dead.”

“You’ve got to be joking.”

“I’m completely serious. Your professor knew things that could have blown up in their faces.”

“Holy shit. And you’re off to Stockholm like some private detective to investigate it all?”

“Not like a private detective, Alona. I’m going to be official, and while I’m there I’m going to give our hacker such a pummelling she won’t be able to stand.”

“Wait, Ed. Did I hear you say she?”

“You’d better believe it. Our hacker’s a she!”

August’s drawings took Salander back in time. She thought of that fist beating rhythmically and relentlessly on the mattress.

She remembered the thuds and the grunting and the crying from the bedroom next door. She remembered the times at Lundagatan when her comics and fantasies of revenge were her only refuge. But she shook off the memories. She changed the dressing on her shoulder. Then she checked her pistol, made sure that it was loaded. She went onto the P.G.P. link. Andrei Zander was asking how they were, and she gave a short reply.

Outside, the storm was shaking the trees and bushes. She helped herself to some whisky and a piece of chocolate, then went out onto the terrace and from there to the rock slope where she carefully reconnoitred the terrain, noticing a small cleft part way down. She counted her steps and memorized the lie of the land.

By the time she got back, August had made another drawing of Westman and the Roger person. She supposed he needed to get it out of his system. But still he had not drawn anything from the night of the murder. Perhaps the experience was blocked in his mind.

Salander was overcome by a feeling of time running away from them and she cast a worried look at August. For a minute or so she focused on the mind-boggling numbers he had put down on paper next to the new drawing. She studied their structure until suddenly she spotted a sequence which did not fit in with the others.

It was relatively short: 2305843008139952128. She got it immediately. It was not a prime number, it was — and here her spirits lifted — a number which, according to a perfect harmony, is made up of the sum of all its positive divisors. It was, in other words, a perfect number, just as 6 is because it can be divided by 3, 2 and 1 and 3 + 2 + 1 happen to add up to 6. She smiled. And then she had an exhilarating thought.

“Now you’re going to have to explain yourself,” Casales said.

“I will,” Needham said. “But first, even though I trust you, I need you to give me a solemn promise that you won’t say any of this to anybody.”

“I promise, you jerk.”

“Good. Here’s the story: after I yelled at Ingram, mostly for the sake of appearances, I told him he was right. I even pretended to be grateful to him for putting a stop to our investigation. We wouldn’t have gotten any further anyway, I said, and it was partly true. From a purely technical point of view we were out of options. We’d done everything and then some, but it was pointless. The hacker put red herrings all over the place and kept leading us into new mazes and labyrinths. One of my guys said that even if we got to the end, against all odds, we wouldn’t believe we’d made it. We’d just kid ourselves that it was a new trap. We were prepared for just about anything from this hacker, anything but flaws and weaknesses. So if we kept going the usual way we’d had it.”

“You don’t tend to go the usual way.”

“No, I prefer the roundabout way. The truth is, we hadn’t given up at all. We’d been talking to our hacker contacts out there and our friends in the software companies. We did advanced searches, surveillance and our own computer breaches. You see, when an attack is as complex as this one, you can always be sure there’s been some research up front. Certain specific questions have been asked. Certain specific sites have been visited and inevitably some of that becomes known to us. But there was one factor above all that played into our hands, Alona: the hacker’s skill. It was so incredible that it limited the number of suspects. Like a criminal suddenly running a hundred metres in 9.7 seconds at a crime scene — you’d be pretty sure the guy is a certain Mr Bolt or one of his close rivals, right?”

“So it’s at that level?”

“Well, there are parts of this attack that just made my jaw drop, and I’ve seen a fair amount in my day. That’s why we spent a hell of a lot of time talking to hackers and insiders in this industry and asking them who is capable of something really, really big? Who are the seriously big players these days? We had to be pretty smart about how we framed our questions, so that nobody would guess what actually happened. For a long time we got nowhere. It was like shooting in the dark — like calling out into the dead of night. Nobody knew anything, or they claimed they didn’t. A few names were mentioned, but none of them felt right. For a while we chased down some Russian, a Jurij Bogdanov — an ex-druggie and thief who apparently can hack into anything he damn well likes. The security companies were already trying to recruit him when he was living on the street in St Petersburg, hot-wiring cars, weighing in at forty kilos of skin and bone. Even the people from the police and intelligence services wanted him on their side. They lost that battle, needless to say. These days Bogdanov looks clean and successful and has ballooned to sixty kilos of skin and bone, but we’re pretty sure he’s one of the crooks in your organization, Alona. That was another reason he interested us. There had to be a connection to the Spiders, because of the searches that got carried out, but then...”

“You couldn’t understand why one of their own would be giving us new leads and associations?”

“Exactly, and so we looked further. After a while another outfit cropped up in the conversations.”

“Which one?”

“They call themselves Hacker Republic. They have a big reputation out there. A bunch of talents at the top of their game and rigorous about their encryptions. And for good reason. We’re constantly trying to infiltrate these groups, and we’re not the only ones. We don’t just want to find out what they’re up to, we also want to recruit their people. These days there’s big competition for the sharpest hackers.”

“Now that we’ve all become criminals.”

“Ha, yes, maybe. Whatever, Hacker Republic has major talent. Lots of the guys we talked to backed that up. And it wasn’t just that. There were also rumours that they had something big going on, and then a hacker with the handle Bob the Dog, who we think is linked to the gang, was running searches and asking questions about one of our guys, Richard Fuller. Do you know him?”

“No.”

“A manic-depressive self-righteous prick who’s been bugging me for a while. The archetypal security risk, who gets arrogant and sloppy when he’s in a manic phase. He’s just the kind of person a bunch of hackers should be targeting, and you’d need classified information to know that. His mental-health issues aren’t exactly common knowledge — his own mother hardly knows. But I’m pretty confident that in the end they didn’t get in via Fuller. We’ve examined every file he’s received recently and there’s nothing there. We’ve scrutinized him from top to bottom. But I bet Fuller was part of Hacker Republic’s original plan and then they changed strategy. I can’t claim to have any hard evidence against them, not at all, but my gut feeling is still that these guys are behind the break-in.”