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Density of pythagorean triplets – Part 2

Several people really liked my previous puzzle, including some high-school students 🙂 Some of you have sent me photos of your hand-written proofs. This level of enthusiasm for mathematics is totally awesome. Here is a detailed proof with some beautiful diagrams by Chris Grossack.

Here is my next puzzle:

Let x_1 < x_2 and a^2 + {x_1}^2 = {y_1}^2 and a^2 + {x_2}^2 = {y_2}^2.

Let x_3 < x_4 and b^2 + {x_3}^2 = {y_3}^2 and b^2 + {x_4}^2 = {y_4}^2.

Also, x_2 - x_1 = x_4 - x_3 = \delta

All the above variables are positive integers and a \neq b and x_1 \neq x_3 and x_2 \neq x_4

Prove or disprove the following claim:

Claim: There exists a rational number 0 < q < \delta such that \sqrt{a^2 + {(x_1 + q)}^2} and \sqrt{b^2 + {(x_3 + q)}^2} are rational numbers.

If you want to take small baby steps towards a proof, start with the following special cases:

Special case 1: x_1 = x_3 and x_2 = x_4

Special case 2: x_1 = x_3 and x_2 = x_4 and a = b

Quick homework problem: Prove the Special case 2 using a proof of my previous puzzle.

Have fun solving.

Some mathematician has said pleasure lies not in discovering truth, but in seeking it 🙂

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Density of pythagorean triplets

I designed this simple math puzzle last week:

Prove or disprove the following statement:

  • Claim: Let p, q, r, s be positive integers such that p < q and r < s. Given two rational numbers \frac{p}{q} < \frac{r}{s}, there exist positive integers a, b such that a < b and \frac{p}{q} < \frac{a}{b} < \frac{r}{s} and a^2 + b^2 is a perfect square.
  • If the above statement is true, show an example of a, b, expressed in terms of p, q, r, s.
  • If the above statement is false, construct a counterexample.

Leave your solutions in the comments. Have fun solving it.

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EulerChain Cryptocurrency Mathematics Bounty 1

I am announcing cryptocurrency bounties for solving some of my favorite problems in mathematics. Here is the first one:

  • Let P=(v1, v2, v3, v4) be a simple polygon drawn on a plane.
  • The co-ordinates of the vertices v1, v2, v3, v4 are all rational numbers.
  • The lengths of the edges (v1, v2), (v2, v3), (v3, v4) and (v4, v1) are all integers.
  • The distance between v1 and v3 is an integer.

Conjecture 1: There exists a point x with rational coordinates inside P such that the euclidean distances between the pairs (x, v1), (x, v2), (x, v3), (x, v4) are all rational numbers.

Conjecture 2: Same as Conjecture 1 when the polygon P is convex.

Eulerchain bounties:

  • 100 Eulercoins for proving Conjecture 1. This implies Conjecture 2 is also true.
  • 50 Eulercoins for disproving only Conjecture 1.
  • 50 Eulercoins for proving only Conjecture 2.
  • 100 Eulercoins for disproving Conjecture 2. This implies Conjecture 1 is also false.
  • The bounties are valid till Dec 31 2021. If they are not resolved by Dec 31 2021, I will revisit this and update the bounties.

If you have any questions (or) solutions (or) counter-examples, leave a comment.

Happy Solving.

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Prove or disprove this geometry conjecture

I am working on a geometry problem and it lead me to make the following conjecture:

Conjecture 1: Given any convex polygon (v1, v2, …. v_n) drawn on a plane with integer edge lengths, there is a point x inside the polygon such that the euclidean distances between the pairs (x, v1), (x, v2), … (x, v_n) are all rational numbers.

Theorem: Conjecture 1 is true for n = 3.

Simple Homework: Prove Conjecture 1 for an equilateral triangle.

Conjecture 1 seems like a very natural geometry problem. Is it well-known ? If you know any references, please leave a comment or email me.

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Anonymous Photo Journalism using freebits

I am very excited to announce ANONYMOUS accounts in the freebits iOS app using decentralized public key cryptography. Username, email and password are NOT required.

Citizen journalists, photo journalists and anyone anywhere in the world can use freebits to prove the authenticity of their photos, videos (captured using our iOS app) without ever revealing their real-world identity.

iOS app: https://apps.apple.com/us/app/freebits/id1582246045

Web app: http://freebits.app

A poll of 1,762 non-satellite TV stations and 3,379 radio outlets reveals journalists were threatened, assaulted, and even arrested at alarming rates across the U.S. in 2020. More details are in this article.

What is freebits ?

Freebits is a radically new kind of social media. It has built-in support for decentralized identities, privacy-preserving anonymous identities, AI-powered news aggregation, tamper-proof photos, videos, credentials and tamper-proof on-chain storage of socio-economic data (eg: unemployment numbers, prices of consumer goods, CPI to prevent data manipulation by governments and other centralized entities).Freebits is algorithm-free i.e., your posts will not be shadow-banned or hidden or suppressed.


Our mission is to enable journalists, creators, influencers, educators, podcasters, artists, politicians, activists, citizen journalists, photo journalists, data journalists, global leaders and everyone to speak freely to everyone globally, connect with their followers and get tips without any fear of censorship or shadow banning or de-platforming.


Freebits version 2.0 supports privacy-preserving anonymous accounts using public key cryptography. Username, email and password are not required. Citizen journalists, photo journalists and anyone anywhere in the world can use freebits to prove the authenticity of their photos, videos (captured using our iOS app) without ever revealing their real-world identity. Every photo and video shot using freebits app is a cryptographically-signed, verifiable and tamper-proof digital asset. The provenance and the digital rights of these assets are securely tracked using our Blockchain.

http://www.freebits.app

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Online version of Freebits White Paper

Online version of freebits white paper is now available as a medium post : https://kintali.medium.com/freebits-white-paper-814d97b81e18


The intention of releasing this paper is to persuade everyone with factual & technical evidence that there is a global problem that must be addressed immediately & our solution is a superior method of solving it.

A pdf file of the white paper was released last month. It is available at http://shivakintali.org/freebits.pdf

Also, ❤️🧡💛💚💙💜 Happy Pride month 🌈 to every global citizen… from the Freebits team.

Love, equality, human rights, free speech and freedom of expression for everyone. 🕺💃🎉🥳🍾

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Join Freebits and get free Euler cryptocurrency

Today we are moving freebits (a decentralized twitter) from “beta” stage to “stable release” stage. Anyone from anywhere can signup and start using all the features.

Join Freebits and get free Euler cryptocurrency, proportional to the number of your twitter followers.

Only for the first 10,000 signups.

No hidden games. No scams. No spammy emails. You create your decentralized ID and own your private keys. You own your Euler. We will never ask you to send your bitcoin or any other crypto. We will never ask you to share your private keys or seed phrases.

For more details, please read my medium post 👉 https://kintali.medium.com/freebits-a-decentralized-global-public-square-89581235948

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Freebits – A decentralized public square

I am very excited to announce Freebits ( http://freebits.app ), a decentralized, permissionless, immutable and uncensorable social network built on top of EulerChain.

Our mission is to enable journalists and influencers to speak freely to everyone globally, connect with their fans and get tips (in crypto) without any censorship or de-platforming.

Please sign up and join the future of social media 👉 https://freebits.app/

What is freebits ? 👉 https://freebits.app/about

FAQ 👉 https://freebits.app/faq/

My profile on freebits 👉 https://freebits.app/shiva/