Euclid
∞A proof that there are infinitely many prime numbers.
AUTHOR NARRATION
Hear the poem.
Read by the author in the original English.
Odd Imperfect FriendEnglish narration
POEM + PRIMES
Odd Imperfect Friend
WHEN THE PARTS EQUAL THE WHOLE
Perfect Numbers
A positive integer is perfect when its proper divisors—its positive factors smaller than itself—add up to the number itself.
A LITTLE NUMBER LABORATORY
Hunt for a perfect number
Choose an example to see how its parts equal the whole. Then try your own integer or draw a large random odd one.
START WITH A KNOWN PERFECT NUMBER
A LITTLE NUMBER LABORATORY
Yes, this is a perfect number!
PROPER DIVISORS
=124714
SUM OF PROPER DIVISORS28THE NUMBER28
If genuine, this would be a major discovery. Please verify it independently before sharing it.
Results are exact for accepted integers. Very large numbers may take a few seconds; if a calculation cannot finish, no verdict is shown.
THE ODD QUESTION
Does an odd perfect number exist?
Hover or focus to reveal what is known.Every perfect number discovered so far is even. No one yet knows whether an odd perfect number exists. Any odd perfect number would have to exceed 101500.Mathematical source →
AN UNSOLVED PROBLEM
Goldbach’s Conjecture
Can every even integer greater than 2 be written as the sum of two prime numbers?
That is the strong Goldbach conjecture. A prime is a whole number greater than 1 divisible only by 1 and itself. The statement is seductively simple, and every even number in this exploration satisfies it, yet no proof covering all even integers is known.
Hover, focus with the keyboard, or tap any even number to see all of its decompositions into two primes.
Read an American Mathematical Society introduction to the conjecture →INTEGERS 1–200
even number — interactiveodd number
123579111315171921232527293133353739414345474951535557596163656769717375777981838587899193959799101103105107109111113115117119121123125127129131133135137139141143145147149151153155157159161163165167169171173175177179181183185187189191193195197199
A CONJECTURE IN APPROACH
The Twin Prime Conjecture
Twin primes, such as 11 and 13, differ by only 2. The conjecture says that infinitely many such pairs exist. For a long time, progress could only produce a gap that still grew with the primes themselves. In 2013, Yitang Zhang crossed a frontier: he proved that one fixed ceiling works infinitely often. That ceiling has since been tightened dramatically. Mathematicians have not yet reached the gap of 2—but the history of getting closer is already remarkable.
FROM INFINITE PRIMES TO RELATIVELY SMALL GAPS
- c. 300 BCE
- 1896
Prime Number Theorem
≈ log pThe average gap near p is on the scale of log p, providing a natural yardstick for prime gaps.
- 2005
Goldston–Pintz–Yıldırım
o(log p)Infinitely many gaps are smaller than every fixed fraction of the average gap—but no constant ceiling yet.
AN ABSOLUTE BOUND — THE TURNING POINT
Apr. 2013Yitang Zhang
≤ 70,000,000The first unconditional proof of bounded gaps: one absolute constant works infinitely often.
- 2013
Polymath8a · Terence Tao
≤ 4,680Tao helped organize an open collaboration that rapidly sharpened Zhang’s method, lowering the bound by more than four orders of magnitude.
- Nov. 2013
James Maynard
≤ 600A new multidimensional sieve made the argument more flexible and tightened the interval again.
- Nov. 19, 2013
Terence Tao’s parallel work
NEW SIEVETao independently arrived at closely related sieve ideas—with a slightly weaker general bound—then launched Polymath8b to combine them with Maynard’s method.
- 2014–present
Polymath8b · launched by Tao
≤ 246The collaboration refined the Maynard–Tao sieve to the best known unconditional bound: infinitely many intervals of length 246 contain at least two primes.
STILL-OPEN TARGET
?Twin primes
= 2Proving this final step would establish that infinitely many twin-prime pairs exist.
A bound ≤ B means that infinitely many consecutive-prime pairs are at most B apart. Because only finitely many even gaps are then possible, at least one fixed gap among them must recur infinitely often—even though we do not yet know which one.
Widths are logarithmic and illustrative: the fall from 70,000,000 to 246 is far more dramatic than a linear scale could display here.
DEGREES OF COLLABORATION
The Erdős Number
The Erdős number works like the Bacon number: where the latter connects performers to Kevin Bacon through shared films, the former connects researchers to Paul Erdős through coauthored papers. Erdős himself has number 0; his direct coauthors have number 1; their coauthors have number 2; and the chain continues outward.
0Paul Erdős
1direct coauthor
2a coauthor’s coauthor
3+and onward
Exact totals vary slightly with database updates and bibliographic criteria: the project archive lists more than 1,400 papers, while biographical sources commonly describe more than 1,500 publications and more than 500 coauthors.
A VOCABULARY—AND LIFE—OF HIS OWN
- “My brain is open”Nearly homeless by choice, he arrived at colleagues’ homes with a small suitcase and this invitation to begin doing mathematics immediately.
- The BookHe imagined an ideal book containing the most beautiful proof of every theorem—a metaphor that became part of mathematical culture.
- EpsilonsIn his private vocabulary, children were “epsilons”; to “leave” meant to die, while to “die” could mean to stop doing mathematics.
- Problems with prizesHe assigned cash rewards to open problems according to their difficulty, turning his conjectures into tangible invitations to collaborate.
- Radical generosityHe gave much of his lecture income and prize money to young mathematicians or recycled it into rewards for new problems.
SOURCES + FURTHER READING
The Erdős Number ProjectMathematical Association of AmericaAmerican Mathematical Society tributeREADER DISCUSSION
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