A problem in the spirit of Fermat’s Last Theorem or the ABC conjecture

If you have k numbers that add up to zero (with possible plus or minus signs), what is the maximum number ‘n’ of prime factors of the number with the least prime factors — what kind of inequality can we say about ‘n’ and ‘k’? The primes can be repeated, as long as there are no common factors for each term.

For example, 2*5*29 + 7*7*11 – 2*7*23 – 3*13*13 == 0 is true. In this case, n = 3 and k = 4. [In this case, the 12 primes are divided into four equal groups of 3, but that is not a requirement in general.]

MOTIVATION: the push and pull of multiplication and addition is an interesting topic. This seems a question in the same vein as Fermat’s Last theorem or the ABC conjecture, and it seems like a very natural similar question to ask. I suspect we’re not even close to being able to answer it, but would like to know if there’s a conjecture regarding ‘n’ and ‘k’, especially as they get larger, similar to the ABC conjecture inequality.

The slow lane paradox, or: you’re not paranoid, the universe really is out to get you

If you are in one of two identical lanes, then for the majority of the time you’ll be in the slower lane.

This fact does not spring from the psychology of human perception, but from mathematics. (The psychological aspect will enthusiastically add to our travails, of course.)

This violates our sense of symmetry, and so can be considered a paradox. The explanation of this paradox is not terribly sophisticated mathematically, although there are some subtleties to do with probability. However, to the best of my knowledge, it hasn’t been described elsewhere; at least, it isn’t familiar to most people. Since this situation comes up all so often, it seems it should be better known.

And it’s a nice little exercise in simple probability, too, so we’ll work it out in detail. Even if the result is obvious, there are some interesting aspects.

Here’s the article:

https://edgeofthecircle.net/lane_paradox.pdf

Faiz Ahmed Faiz: a translation of “Nisaar mein teri galiyo ke”

Faiz Ahmed Faiz’s poem, Nisaar mein teri galiyo ke, is one of my favorite poems. It has been on my mind a lot lately, for whatever reason. I ended up making my own rather loose translation. Sharing the translation first, then the original Urdu poem.

_____________________

A salute and a half to your alleyways, oh gracious land of mine,
Home to the latest fad, that every head remain bowed!
Any discontent, any seeker who dares to venture about
May only slink through the streets with a tremble in their heart.
Those guilty of honesty soon learn the logic of the insane:
Attack dogs may roam free, only the bones of the earth are chained.

Those who deal out oppression need no more pretext
Than the handful of faithful who still whisper your name.
The lords of the land play both accuser and judge,
So who will defend you, from whom will justice flow?
But, for those who yet persist, the time yet drips away.
I’m absent now but listen, my love: these days will slip away.

When the door of the jail grows dark, it’s my heart that can see
How the stars of the sky are now scattered in your mane.
When the chains that bind me are lit, what I perceive
Is the glow of dawn upon your face.
Prison’s days are but mirages of sun’s rises and its falls.
My life crawls captive to the shadows of the walls.

Oh, this is the way of the war of the tyrants and the people.
There’s nothing novel in their rites, nothing novel in our fight.
We can coax roses to bloom from the very midst of the blaze,
So their loss will come soon, and our successes no surprise.
Why should I complain about a little time in the dark?
In this short moment I’m away, I won’t shadow my heart.

So this day we’re apart, the morrow we’re together.
A parting of but one night is no matter at all.
If the oppressor’s fist is now high, it’s really no bother.
This silly half-week of their might is no matter at all.
Those who remember their truths, who hang fast to their reasons,
Hold the power to quell these petty whirlpools of seasons.

Original:
Nisaar mein teri galiyon ke ai watan ke jahan,
Chali hai rasm ke koi na sar utha ke chale,
Jo koi chahne waala tawaaf ko nikle,
Nazar chura ke chale, jism-o-jaan bacha ke chale,
Hai ahl-e-dil ke liye ab ye nazm-e-bast-o-kushaad,
Ke sang-o-khisht muqayyad hain aur sag aazaad.


Bahut hai zulm ke dast-e-bahaana-joo ke liye,
Jo chand ahl-e-junoon tere naam lewa hain,
Bane hain ahl-e-hawas mudd’ai bhi, munsif bhi,
Kise wakeel karein, kisse munsifi chahein,
Magar guzaarne waalon ke din guzarte hain,
Tere firaaq mein yoon subh-o-shaam karte hain.


Bujha jo rozan-e-zindaan to dil ye samjha hai,
Ke teri maang sitaaron se bhar gayi hogi,
Chamak uthe hain salaasil to hum ne jaana hai,
Ke ab seher tere rukh par bikhar gayi hogi,
Gharaz tasavvur-e-shaam-o-seher mein jeete hain,
Giraft-e-saaya-e-deewaar-o-dar mein jeete hain.


Yoon hi hamesha ulajhti rahi hai zulm se khalq,
Na unki rasm nayi hai, na apni reet nayi,
Yoon hi hamesha khilaaye hain hum ne aag mein phool,
Na unki haar nayi hai, na apni jeet nayi,
Isi sabab se falak ka gila nahi karte,
Tere firaaq mein hum dil bura nahi karte.


Gar aaj tujh se juda hain to kal behem honge,
Ye raat bhar ki judaai to koi baat nahi,
Gar aaj auj pe hai taala’-e-raqeeb to kya,
Ye chaar din ki khudai to koi baat nahi,
Jo tujh se ehd-e-wafa ustuwaar rakhte hain,
Ilaaj-e-gardish-e-lail-o-nihaar rakhte hain!

Living Mathematics — a book of math

Here’s a link to the book: https://edgeofthecircle.net/living_mathematics.pdf

This book is intended for two types of people: those who love mathematics, and those who don’t. If you belong to either one of these groups, I hope you will read on.

The mathematics that most people encounter (high school, typically) isn’t a reflection of what higher math is like. This means that people decide whether or not they like mathematics, without ever knowing what mathematics is like.

This book is a journey along the paths of mathematics as it can be. It’s aimed at someone with an approximately high school level of knowledge (or you could be a really motivated middle schooler, that’s okay too!) , with no knowledge of calculus, matrices, or complex numbers assumed.

At the same time, I wanted this to be a book of mathematics, rather than a book about mathematics. What does that mean? You’re not going to be reading about concepts and proofs that other people did, you’re going to be actually taking the steps yourself. And yet, we’re going to actually reach some pretty sophisticated destinations.

To use mountaineering as a metaphor: this book is a hike to the top, as opposed to a technical rock-face climb. It’s easier and doesn’t need as much training, but you’re still going to take the steps yourself to get to the top. And you get to enjoy the view at the end that you arrived at by yourself, rather than look at a photo that someone else took.

https://edgeofthecircle.net/living_mathematics.pdf