Sets of integers with no large sum-free subset Eberhard et al, Ann. Math. 2014.

上次讀過的 Erdos 1965 年的論文裡的幾個問題其中之一，這裡有最新的結果。這裡證明了，對任何的 ε>0，都存在一個整數的子集合 A，|A|=n，讓你從 A 裡隨便取 (1/3 + ε)n 個數字，一定會遇到裡頭有 x、y、z 相異整數，讓…

What is the biggest whole number that you can write down or describe uniquely? Well, there isn’t one, if we allow ourselves to idealize a bit. Just write down “1”, then “2”, then… you’ll never find a last one.

Of course, in real life you’ll die before you get to any…

I know, I know, you may say I’m kind of mad or anything for stating something like that in the title of this text. Actually, if there was no asterisk at the end, we could still have a very nice…

2 — the last digit is even (0, 2, 4, 6, 8) E.G. 128 = 64.

3 — the sum of all digits can be divisible by 3. E.G. 381 (3+8+1 = 12; 12÷3 = 4.

4 — the last 2 digits are divisible by 4. E.G. 268 (68÷4= 17）

In MIT’s Big Number Duel, the rule simply states:

it would be considered unsporting to win by adding one to your opponent’s last entry

Very happy to see that the L-functions and Modular Forms Database is now live!

When I was a kid we looked up our elliptic curves in Cremona’s tables, on paper. Then William Stein created the Modular Forms Database (you can still go there but it doesn’t really work) and suddenly you could…

There’s an idea! Come to think of it, I might be able find a two octave portable keyboard with headphones. It would be a nice change of pace.