GITNUX REPORT 2024

# Decoding the Most Hardest Math Problems in Modern Mathematics

Unsolved for centuries, delve into the enigmatic world of the hardest math problems ever.

Author: Jannik Lindner

First published: 7/17/2024

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The ABC Conjecture was proposed in 1985

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The ABC Conjecture relates the prime factors of the sum of two numbers to the prime factors of those numbers

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A proof of the ABC Conjecture was claimed by Shinichi Mochizuki in 2012, but it remains unverified

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The ABC Conjecture has implications for many other problems in number theory

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If proven, the ABC Conjecture would imply a new proof of Fermat's Last Theorem

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The Birch and Swinnerton-Dyer Conjecture has been open since 1965

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The Birch and Swinnerton-Dyer Conjecture relates to elliptic curves and their L-functions

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A solution to the Birch and Swinnerton-Dyer Conjecture could have applications in cryptography

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The Birch and Swinnerton-Dyer Conjecture is one of the seven Millennium Prize Problems

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Partial results on the Birch and Swinnerton-Dyer Conjecture have been proven for elliptic curves of rank 0 and 1

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The Collatz Conjecture was first proposed in 1937

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The Collatz Conjecture has been verified for all starting numbers up to 2^68

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Paul Erdős said about the Collatz Conjecture: 'Mathematics may not be ready for such problems'

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The Collatz Conjecture is also known as the 3n + 1 problem

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A \$500 prize is offered for resolving the Collatz Conjecture

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The Goldbach Conjecture was proposed in 1742

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The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two primes

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The weak Goldbach Conjecture (every odd number greater than 5 is the sum of three primes) was proven in 2013

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The Goldbach Conjecture has been verified for all even numbers up to at least 4 × 10^18

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The Goldbach Conjecture is one of the oldest unsolved problems in number theory and mathematics

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The Hodge Conjecture was formulated in 1950

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The Hodge Conjecture relates algebraic geometry and differential geometry

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The Hodge Conjecture is one of the seven Millennium Prize Problems

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A special case of the Hodge Conjecture for abelian varieties was proven in 2006

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The Hodge Conjecture has implications for the study of complex manifolds

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The Jacobian Conjecture was first formulated in 1939

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The Jacobian Conjecture remains unsolved despite numerous attempts

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A \$10,000 prize is offered for resolving the Jacobian Conjecture

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The Jacobian Conjecture has been proven for polynomials of degree 2

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The Jacobian Conjecture is related to the study of polynomial automorphisms

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The Langlands Program was proposed in the 1960s

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The Langlands Program is often described as a 'grand unified theory' of mathematics

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The Langlands Program connects number theory, algebraic geometry, and representation theory

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Significant progress on the Langlands Program was made by Laurent Lafforgue, earning him a Fields Medal in 2002

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The Langlands Program is considered one of the most ambitious projects in modern mathematics

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The Navier-Stokes Existence and Smoothness problem was formulated in the 19th century

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The Navier-Stokes Equations describe the motion of fluids

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A solution to the Navier-Stokes problem could have applications in weather prediction and aerodynamics

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The Navier-Stokes Equations are one of the seven Millennium Prize Problems

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Partial results on the Navier-Stokes problem have been obtained for simplified versions of the equations

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The Hadwiger-Nelson problem, concerning the chromatic number of the plane, has been open since 1950

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The Invariant Subspace Problem for Hilbert spaces has been open since 1935

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The Erdős discrepancy problem, proposed in the 1930s, was solved in 2015 using computer-assisted proof

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The Kissing Number Problem in dimensions 8 and 24 was solved in 2003 and 2004, respectively

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The Kepler Conjecture, proposed in 1611, was finally proved in 2014 using computer-assisted techniques

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The P vs NP problem has been open since 1971

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The P vs NP problem is considered one of the most important open problems in computer science

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A solution to the P vs NP problem could have profound implications for cryptography

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The P vs NP problem is one of the seven Millennium Prize Problems

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Over 100 incorrect proofs of P ≠ NP have been published

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The Poincaré Conjecture was formulated in 1904

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The Poincaré Conjecture was proven by Grigori Perelman in 2002-2003

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Perelman declined the \$1 million Millennium Prize for solving the Poincaré Conjecture

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The proof of the Poincaré Conjecture took over 100 years to complete

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The Poincaré Conjecture is the only Millennium Problem to have been solved so far

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The Riemann Hypothesis has remained unsolved for over 160 years

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The Riemann Hypothesis is one of the seven Millennium Prize Problems

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A proof of the Riemann Hypothesis would have significant implications for the distribution of prime numbers

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The Clay Mathematics Institute offers a \$1 million prize for solving the Riemann Hypothesis

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Over 10,000 papers have been published on the Riemann Hypothesis

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The Twin Prime Conjecture was first stated in 1849

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The Twin Prime Conjecture states that there are infinitely many pairs of primes that differ by 2

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In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million

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The bound for the gap between twin primes has been reduced to 246 as of 2019

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The Twin Prime Conjecture is related to the broader study of the distribution of prime numbers

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## Summary

• The Riemann Hypothesis has remained unsolved for over 160 years
• The Riemann Hypothesis is one of the seven Millennium Prize Problems
• A proof of the Riemann Hypothesis would have significant implications for the distribution of prime numbers
• The Clay Mathematics Institute offers a \$1 million prize for solving the Riemann Hypothesis
• Over 10,000 papers have been published on the Riemann Hypothesis
• The Birch and Swinnerton-Dyer Conjecture has been open since 1965
• The Birch and Swinnerton-Dyer Conjecture relates to elliptic curves and their L-functions
• A solution to the Birch and Swinnerton-Dyer Conjecture could have applications in cryptography
• The Birch and Swinnerton-Dyer Conjecture is one of the seven Millennium Prize Problems
• Partial results on the Birch and Swinnerton-Dyer Conjecture have been proven for elliptic curves of rank 0 and 1
• The P vs NP problem has been open since 1971
• The P vs NP problem is considered one of the most important open problems in computer science
• A solution to the P vs NP problem could have profound implications for cryptography
• The P vs NP problem is one of the seven Millennium Prize Problems
• Over 100 incorrect proofs of P ≠ NP have been published

Diving into the realm where numbers dance and minds wrestle, some mathematical problems seem to have taken a vow of eternal mystery, teasing scholars for centuries like elusive phantoms in the night. From the enigmatic whispers of the Riemann Hypothesis to the tantalizing allure of the Twin Prime Conjecture, these mathematical conundrums have perplexed the brightest minds and ignited a fiery passion for discovery. With over 10,000 papers devoted to unraveling their enigmas, these problems stand as the ultimate testament to the insatiable curiosity and unyielding determination of the human intellect. Welcome to the heart of the Most Hardest Math Problem jungle, where solutions are worth a million-dollar chase and where even the gods of numbers tread lightly.

## ABC Conjecture

• The ABC Conjecture was proposed in 1985
• The ABC Conjecture relates the prime factors of the sum of two numbers to the prime factors of those numbers
• A proof of the ABC Conjecture was claimed by Shinichi Mochizuki in 2012, but it remains unverified
• The ABC Conjecture has implications for many other problems in number theory
• If proven, the ABC Conjecture would imply a new proof of Fermat's Last Theorem

### Interpretation

The ABC Conjecture is the elusive unicorn of mathematics, captivating the minds of number theorists as they strive to unlock its enigmatic secrets. Proposed over three decades ago, its tantalizing promise to unveil the hidden dance of prime factors in number summation continues to baffle and intrigue. Despite the bold claim of proof by the enigmatic Shinichi Mochizuki, the mathematical community remains in a state of suspense, eagerly awaiting confirmation of this mathematical holy grail. If the ABC Conjecture were to unveil its mysteries, it would not only revolutionize number theory but also pave the way for a new victory lap around Fermat's Last Theorem. Until then, mathematicians worldwide remain on the edge of their seats, eagerly anticipating the day when this tantalizing enigma is finally unveiled.

## Birch and Swinnerton-Dyer Conjecture

• The Birch and Swinnerton-Dyer Conjecture has been open since 1965
• The Birch and Swinnerton-Dyer Conjecture relates to elliptic curves and their L-functions
• A solution to the Birch and Swinnerton-Dyer Conjecture could have applications in cryptography
• The Birch and Swinnerton-Dyer Conjecture is one of the seven Millennium Prize Problems
• Partial results on the Birch and Swinnerton-Dyer Conjecture have been proven for elliptic curves of rank 0 and 1

### Interpretation

The Birch and Swinnerton-Dyer Conjecture is like the enigmatic cousin at the family gathering who always leaves you intrigued but unsure of what they're up to. Since 1965, mathematicians have been scratching their heads over the relationship between elliptic curves and their L-functions, with the promise of unlocking secrets that could revolutionize cryptography. Being one of the elite Seven Millennium Prize Problems, it's the kind of brain teaser that separates the mathletes from the math-mehs. While progress has been made on elliptic curves with ranks 0 and 1, the ultimate solution to this tantalizing mystery still remains beyond the grasp of even the most nimble number-crunchers.

## Collatz Conjecture

• The Collatz Conjecture was first proposed in 1937
• The Collatz Conjecture has been verified for all starting numbers up to 2^68
• Paul Erdős said about the Collatz Conjecture: 'Mathematics may not be ready for such problems'
• The Collatz Conjecture is also known as the 3n + 1 problem
• A \$500 prize is offered for resolving the Collatz Conjecture

### Interpretation

The Collatz Conjecture, also humorously known as the 3n + 1 problem, has been tantalizing mathematicians since 1937 with its deceptively simple yet stubbornly unsolved nature. Despite reaching mammoth heights with verification up to 2^68, even legendary mathematician Paul Erdős wryly suggested that perhaps the world of numbers was not quite prepared for the headache-inducing intricacies the Collatz Conjecture presents. With a humble \$500 prize dangling as both a lure and a testament to the sheer challenge it poses, this mathematical enigma continues to defy resolution, daring the boldest minds to grapple with its elusive secrets.

## Goldbach Conjecture

• The Goldbach Conjecture was proposed in 1742
• The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two primes
• The weak Goldbach Conjecture (every odd number greater than 5 is the sum of three primes) was proven in 2013
• The Goldbach Conjecture has been verified for all even numbers up to at least 4 × 10^18
• The Goldbach Conjecture is one of the oldest unsolved problems in number theory and mathematics

### Interpretation

The Goldbach Conjecture, a puzzle as timeless as the pyramids but without the luxurious accommodations, has baffled mathematicians for centuries with its tantalizing promise of simplicity hidden within the labyrinth of numbers. Like a mathematically inclined Sherlock Holmes, it challenges us to unravel the enigmatic connections between primes and even integers, teasing us with the allure of a definitive solution just beyond our grasp. While we may have cracked the code for odd numbers (thanks, 2013!), the realm of even integers remains a tantalizing mist, obscuring the ultimate truth. Will we one day unlock the secrets hidden within the Goldbach Conjecture, or will its mystery persist like an elusive mathematical specter haunting the halls of academia? The numbers may never tell, but the thrill of the chase persists unabated.

## Hodge Conjecture

• The Hodge Conjecture was formulated in 1950
• The Hodge Conjecture relates algebraic geometry and differential geometry
• The Hodge Conjecture is one of the seven Millennium Prize Problems
• A special case of the Hodge Conjecture for abelian varieties was proven in 2006
• The Hodge Conjecture has implications for the study of complex manifolds

### Interpretation

The Hodge Conjecture, like a timeless celebrity romance, has been captivating mathematicians since its debut in 1950, weaving together the intricate threads of algebraic and differential geometry. As one of the coveted seven Millennium Prize Problems, this enigmatic enigma has managed to maintain an air of mystery, with only a special case for abelian varieties being captured in the spotlight in 2006. Like a complex yet captivating riddle, the Hodge Conjecture beckons mathematicians to unravel its secrets, promising profound insights into the intricate world of complex manifolds.

## Jacobian Conjecture

• The Jacobian Conjecture was first formulated in 1939
• The Jacobian Conjecture remains unsolved despite numerous attempts
• A \$10,000 prize is offered for resolving the Jacobian Conjecture
• The Jacobian Conjecture has been proven for polynomials of degree 2
• The Jacobian Conjecture is related to the study of polynomial automorphisms

### Interpretation

The enigmatic Jacobian Conjecture is like that elusive puzzle piece teasing mathematicians for decades. Its stubborn refusal to be cracked has turned it into the holy grail of mathematical conundrums, with a tempting \$10,000 bounty dangling like a carrot in front of eager problem-solvers. Though the Conjecture surrenders for polynomials of the second degree, its defiance lingers on, entangled in the intricate web of polynomial automorphisms. Like an unsolved riddle whispered by numbers, the Jacobian Conjecture dares mathematicians to dance with complexity, leaving them tantalizingly close yet frustratingly distant from its ultimate solution.

## Langlands Program

• The Langlands Program was proposed in the 1960s
• The Langlands Program is often described as a 'grand unified theory' of mathematics
• The Langlands Program connects number theory, algebraic geometry, and representation theory
• Significant progress on the Langlands Program was made by Laurent Lafforgue, earning him a Fields Medal in 2002
• The Langlands Program is considered one of the most ambitious projects in modern mathematics

### Interpretation

The Langlands Program, like a mathematical wizard's spellbook, weaves together the mystical realms of number theory, algebraic geometry, and representation theory into a 'grand unified theory' that has sent mathematicians on an epic quest since the 1960s. Laurent Lafforgue, a modern-day Merlin, made significant strides in deciphering its enigmatic codes, earning the mathematical equivalent of a knight's accolade with the Fields Medal in 2002. This noble quest, dubbed one of the most ambitious projects in modern mathematics, continues to challenge and inspire scholars to unlock the secrets of the numbers' universe.

## Navier-Stokes Equations

• The Navier-Stokes Existence and Smoothness problem was formulated in the 19th century
• The Navier-Stokes Equations describe the motion of fluids
• A solution to the Navier-Stokes problem could have applications in weather prediction and aerodynamics
• The Navier-Stokes Equations are one of the seven Millennium Prize Problems
• Partial results on the Navier-Stokes problem have been obtained for simplified versions of the equations

### Interpretation

The Navier-Stokes Existence and Smoothness problem has been lingering in the mathematical realm like a ghost from the 19th century, haunting fluid dynamics enthusiasts with its tantalizing potential applications in weather forecasting and aerodynamics. As one of the seven Millennium Prize Problems, it stands as a formidable enigma, drawing curious minds into its intricate web of equations. While partial results have been unearthed in the pursuit of solving this elusive riddle, the Navier-Stokes Equations continue to ripple through the scientific community, challenging mathematicians to dive deeper into the turbulent waters of complexity. It's a mathematical saga fit for the ages, with more twists and turns than a river in flood.

## Other Hard Problems

• The Hadwiger-Nelson problem, concerning the chromatic number of the plane, has been open since 1950
• The Invariant Subspace Problem for Hilbert spaces has been open since 1935
• The Erdős discrepancy problem, proposed in the 1930s, was solved in 2015 using computer-assisted proof
• The Kissing Number Problem in dimensions 8 and 24 was solved in 2003 and 2004, respectively
• The Kepler Conjecture, proposed in 1611, was finally proved in 2014 using computer-assisted techniques

### Interpretation

In the world of mathematics, some problems age like fine wine, gaining complexity and allure over the decades. From the enigmatic allure of the Hadwiger-Nelson conundrum, to the timeless allure of the Kepler Conjecture, these mathematical puzzles have been testing the limits of human intellect for generations. In a delightful display of perseverance and computational prowess, solutions have finally emerged for some, reminding us that even the most stubborn mathematical mysteries can be cracked with a little bit of ingenuity - and a lot of computer crunching power. The moral of the story? Keep calm and math on, because even the most stubborn problems can eventually succumb to the relentless march of human knowledge.

## P vs NP Problem

• The P vs NP problem has been open since 1971
• The P vs NP problem is considered one of the most important open problems in computer science
• A solution to the P vs NP problem could have profound implications for cryptography
• The P vs NP problem is one of the seven Millennium Prize Problems
• Over 100 incorrect proofs of P ≠ NP have been published

### Interpretation

The P vs NP problem is the mathematical equivalent of a celebrity feud that has been captivating the computer science world since 1971. Like the elusive unicorn of mathematics, a solution to this problem could unlock a treasure trove of secrets in the realm of cryptography. With over 100 incorrect proofs floating around, it seems everyone wants a piece of the P vs NP drama, but only time will tell if this mathematical soap opera will have a Hollywood ending.

## Poincaré Conjecture

• The Poincaré Conjecture was formulated in 1904
• The Poincaré Conjecture was proven by Grigori Perelman in 2002-2003
• Perelman declined the \$1 million Millennium Prize for solving the Poincaré Conjecture
• The proof of the Poincaré Conjecture took over 100 years to complete
• The Poincaré Conjecture is the only Millennium Problem to have been solved so far

### Interpretation

The tale of the Poincaré Conjecture reads like a mathematical soap opera with more twists and turns than a rollercoaster. From its inception in 1904 to its groundbreaking proof by Grigori Perelman in the early 2000s, the journey of this elusive problem is a testament to the never-ending quest for knowledge. Perelman's shocking refusal of the \$1 million prize only added another layer of intrigue to the saga, leaving mathematicians scratching their heads and the rest of us mortals wondering what makes a genius tick. With the proof taking over a century to complete and the Poincaré Conjecture standing tall as the lone Millennium Problem conquered, one thing is crystal clear - when it comes to math, patience truly is a virtue.

## Riemann Hypothesis

• The Riemann Hypothesis has remained unsolved for over 160 years
• The Riemann Hypothesis is one of the seven Millennium Prize Problems
• A proof of the Riemann Hypothesis would have significant implications for the distribution of prime numbers
• The Clay Mathematics Institute offers a \$1 million prize for solving the Riemann Hypothesis
• Over 10,000 papers have been published on the Riemann Hypothesis

### Interpretation

The Riemann Hypothesis stands as the cryptic enigma that has taunted mathematicians for over 160 years, tempting them with the allure of the elusive solution and the dangling carrot of a \$1 million prize from the Clay Mathematics Institute. As over 10,000 papers have been penned in pursuit of its resolution, one cannot help but wonder if this mathematical conundrum is merely a cosmic joke, designed to humble even the most formidable minds in the realm of numbers. If cracked, the Riemann Hypothesis could unlock the secrets of prime numbers and forever alter the landscape of mathematics, proving once and for all that even the most daunting challenges are nothing more than opportunities for the curious to unravel the mysteries of the universe.

## Twin Prime Conjecture

• The Twin Prime Conjecture was first stated in 1849
• The Twin Prime Conjecture states that there are infinitely many pairs of primes that differ by 2
• In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million
• The bound for the gap between twin primes has been reduced to 246 as of 2019
• The Twin Prime Conjecture is related to the broader study of the distribution of prime numbers

### Interpretation

The Twin Prime Conjecture is like a math mystery novel that keeps readers on the edge of their seats, with plot twists and revelations happening over centuries. From the initial bold claim in 1849 to Yitang Zhang's game-changing breakthrough in 2013, the journey to uncover the elusive twin primes has been nothing short of a mathematical rollercoaster. As mathematicians continue to chip away at the gap between these elusive pairs, one thing is for sure: the pursuit of prime numbers is not just about finding patterns, but about unraveling the secrets of the universe's most enigmatic numerical codes.