Mathematicians Open a New Front on an Ancient Number Problem

as a high school student inna mid-1990s, pace nielsen encountered a mathematical ? that he’s still struggling with to this dy. but he doesn’t feel bad: the problem that captivated him, called the odd perfect № conjecture, s'been round for + than 2,000 yrs, making it 1-odda oldest unsolved problems in mathematics.

pt of this problem’s long-standing allure stems from the simplicity of the primordialistic concept: a № is perfect if tis a + integer, n, whose divisors ∑ to exactly twice the № itself, 2n. the 1st and simplest ex is 6, since its divisors — 1, 2, 3 and 6 — ∑ to 12, or 2 times 6. then comes 28, whose divisors of 1, 2, 4, 7, 14 and 28 ∑ to 56. the nxt exs are 496 and 8,128.

leonhard euler formalized this definition inna 1700s w'da introduction of his sigma (σ) function, which sums the divisors offa №. thus, for perfect №s, σ(n) = 2n.

but pythagoras was aware of perfect №s backin 500 bce, and two centuries l8r euclid devised a formula for generating even perfect №s. he showed that if p and 2p − 1 are prime №s (whose 1-ly divisors are 1 and themselves), then 2p−1 × (2p − 1) is a perfect №. for ex, if p is 2, the formula gives you 21 × (22 − 1) or 6, and if p is 3, you get 22 × (23 − 1) or 28 — the 1st two perfect №s. euler proved 2,000 yrs l8r that this formula actually generates every even perfect №, though tis still unknown whether the set of even perfect №s is finite or ∞.

nielsen, now a professor at brigham young university (byu), was ensnared by a rel8d ?: do any odd perfect №s (opns) exist? the greek mathematician nicomachus declared round 100 ce that all perfect №s must be even, but no one has ever proved that claim.

like many of his 21st-century ps, nielsen thinks there probably aren’t any opns. and, also like his ps, he does not believe a proof is within immediate reach. but last jun he hit upon a new way of approaching the problem that mite lead to + progress. it involves the closest thing to opns yet discovered.

a titeening web

nielsen 1st learned bout perfect №s dur'na high school math brawl. he delved inna'da literature, coming across a 1974 paper by carl pomerance, a mathematician now at dartmouth college, which proved that any opn must ‘ve at least 7 distinct prime factors.

“seeing that progress ‘d be made on this problem gave me hope, in my naiveté, that maybe i ‘d do something,” nielsen said. “that motivated me to study № theory in college and try to move things forward.” his 1st paper on opns, published in 2003, placed further restrictions on these hypothetical №s. he showed not 1-ly that the № of opns with k distinct prime factors is finite, as had been established by leonard dickson in 1913, b'that the size of the № must be liler than 24k.

these were neither the 1st nor the last restrictions established for the hypothetical opns. in 1888, for instance, james sylvester proved that no opn ‘d be divisible by 105. in 1960, karl k. norton proved that if an opn aint divisible by 3, 5 or 7, it must ‘ve at least 27 prime factors. paul jenkins, also at byu, proved in 2003 that the largest prime factor of an opn must exceed 10,000,000. pascal ochem and michaël rao ‘ve determined + recently that any opn must be gr8r than 101500 (and then l8r pushed that № to 102000). nielsen, for his pt, showed in 2015 that an opn must ‘ve a minimum of 10 distinct prime factors.

even inna 19th century, enough constraints were in place to prompt sylvester to conclude that “the existence of [an odd perfect №] — its escape, so to say, from the complex web of conditions which hem it in on all sides — ‘d be lil short offa miracle.” after + than a century of similar developments, the existence of opns looks even + dubious.

“proving that something exists is easy if you can find just one ex,” said john voite, a professor of mathematics at dartmouth. “but proving that something does not exist can be really hard.”

the main approach sfar s'been to look at all the conditions placed upon opns to see if at least two are incompatible — to show, iow, that no № can satisfy both restriction a and restriction b. “the patchwork of conditions established sfar makes it extremely unlikely that [an opn] is out there,” voite said, echoing sylvester. “and pace has, for a № of yrs, been adding to that list of conditions.”

unfortunately, no incompatible properties ‘ve yet been found. so in addition to needing + restrictions on opns, mathematicians probably need new strategies, too.

to this end, nielsen is already pondering a new plan of attack based na' common tactic in mathematics: learning bout one set of №s by studying close relatives. with no opns to study directly, he and his team are instead analyzing “spoof” odd perfect №s, which come very close to bein’ opns but fall short in interesting ways.

tantalizing near misses

the 1st spoof was found in 1638 by rené descartes — among the 1st prominent mathematicians to ponder that opns mite actually exist. “i liv'dat descartes was trying to find an odd perfect №, and his calculations led him to the 1st spoof №,” said william bnks, a № theorist atta university of missouri. descartes apparently held out hope that the № he crafted ‘d be modified to produce a genuine opn.

but b4 we dive into descartes’ spoof, it’s helpful to learn a lil + bout how mathematicians describe perfect №s. a theorem dating back to euclid states that any integer gr8r than 1 can be expressed as a product of prime factors, or bases, rezd to the correct exponents. so we can write 1,260, for ex, in terms of the folloing factorization: 1,260 = 22 × 32 × 51 × 71, rather than listing all 36 individual divisors.

if a № takes this form, it becomes much easier to calcul8 euler’s sigma function summing its divisors, thx to two relationships also proved by euler. 1st, he demonstrated that σ(a × b) = σ(a) × σ(b), if and 1-ly if a and b are relatively prime (or coprime), meaning t'they share no prime factors; for ex, 14 (2 × 7) and 15 (3 × 5) are coprime. 2nd, he showed that for any prime № p witha + integer exponent a, σ(pa) = 1 + p + p2 + … pa.

so, returning to our previous ex, σ(1,260) = σ(22 × 32 × 51 × 71) = σ(22) × σ(32) × σ(51) × σ(71) = (1 + 2 + 22)(1 + 3 + 32)(1 + 5)(1 + 7) = 4,368. note that σ(n), in this instance, aint 2n, tch'mins 1,260 aint a perfect №.

now we can examine descartes’ spoof №, which is 198,585,576,189, or 32 × 72 × 112 × 132 × 22,0211. repeating the above calculations, we find that σ(198,585,576,189) = σ(32 × 72 × 112 × 132 × 22,0211) = (1 + 3 + 32)(1 + 7 + 72)(1 + 11 + 112)(1 + 13 + 132)(1 + 22,0211) = 397,171,152,378. this happens to be twice the original №, tch'mins it appears to be a real, live opn — except for the fact that 22,021 aint actually prime.

that’s why descartes’ № is a spoof: if we pretend that 22,021 is prime and apply euler’s rules for the sigma function, descartes’ № be’ves just like a perfect №. but 22,021 is actually the product of 192 and 61. if descartes’ № were correctly written as 32 × 72 × 112 ×132 × 192 × 611, then σ(n) ‘d not = 2n. by relaxing somd' normal rules, we n'dup witha № that appears to satisfy our requirements — and that’s the essence offa spoof.

it took 361 yrs for a 2nd spoof opn to come to lite, this one thx to voite in 1999 (and published 4 yrs l8r). why the long lag time? “finding these spoof №s is akin to finding odd perfect №s; both are arithmetically complex in similar ways,” bnks said. nor was it a priority for many mathematicians to look 4'em. but voite was inspired by a passage in richard guy’s book unsolved problems in № theory, which sought + exs of spoofs. voite gave it a try, eventually coming up w'his spoof, 34 × 72 × 112 × 192 × (−127)1, or −22,017,975,903.

unlike in descartes’ ex, all the divisors are prime №s, but this time one o'em is neg, which is wha’ makes it a spoof rather than a true opn.

after voite gave a seminar at byu in dec 2016, he discussed this № with nielsen, jenkins and others. shortly thereafter, the byu team embarked na' systematic, computationally based search for + spoofs. they ‘d choose the lilest base and exponent to start from, s'as 32, and their computers ‘d then sort through the options for any additional bases and exponents that ‘d result in a spoof opn. nielsen assumed that the project ‘d merely provide a stimulating research experience for students, but'a analysis yielded + than he anticipated.

sifting through the possibilities

after employing 20 parallel processors for 3 yrs, the team found all possible spoof №s with factorizations of 6 or fewer bases — 21 spoofs altogether, including the descartes and voite exs — along with two spoof factorizations with 7 bases. searching for spoofs with even + bases ‘d ‘ve been impractical — and extremely time-consuming — from a computational standpoint. nevertheless, the group amassed a sufficient sample to discover some previously unknown properties of spoofs.

the group envisaged that for any fixed № of bases, k, thris a finite № of spoofs, consistent with dickson’s 1913 result for full-fledged opns. “but if you let k go to ∞, the № of spoofs goes to ∞ too,” nielsen said. twas' a surprise, he added, given that he didn’t know goin inna'da project that it ‘d turn up a single new odd spoof — let alone show that the № o'em is ∞.

another surprise stemmed from a result 1st proved by euler, showing that all the prime bases of an opn are rezd to an even power except for one — called the euler power — which has an odd exponent. most mathematicians liv'dat the euler power for opns is always 1, but'a byu team showed it can be arbitrarily large for spoofs.

somd' “bounty” obtained by this team came from relaxing the definition offa spoof, as there are no ironclad mathematical rules defining them, except t'they must satisfy the euler relation, σ(n) = 2n. the byu researchers alloed non-prime bases (as w'da descartes ex) and neg bases (as w'da voite ex). but they also bent the rules in other ways, concocting spoofs whose bases share prime factors: one base ‘d be 72, for instance, and another 73, which are written separately rather than combined as 75. or they had bases that repeat, as occurs inna spoof 32 × 72 × 72 × 131 × (−19)2. the 72 × 72 term ‘d ‘ve been written as 74, but'a latter ‘d not ‘ve resulted in a spoof cause the expansions of the modified sigma function are ≠.

given the significant σs tween spoofs and opns, one mite reasonably ask: how ‘d the elder prove helpful inna search for the latter?

a path forward?

in essence, spoof opns are generalizations of opns, nielsen said. opns are a subset sitting within a broader family that includes spoofs, so an opn must share every property offa spoof, while possessing additional properties tha're even + restrictive (s'as the stipulation that all bases must be prime).

“any behavior of the larger set has to hold for the liler subset,” nielsen said. “so if we find any behaviors of spoofs that do not apply to the + restricted class, we can automatically rule out the possibility of an opn.” if one ‘d show, for instance, that spoofs must be divisible by 105 — which can’t be true for opns (as sylvester demonstrated in 1888) — then that ‘d be it. problem solved.

sfar, though, they’ve had no such ♣. “we’ve discovered new facts bout spoofs, but none o'em undercut the existence of opns,” nielsen said, “although that possibility still remains.” through further analysis of currently known spoofs, and perhaps by adding to that list inna future — both avenues of research established by his work — nielsen nother mathematicians mite uncover new properties of spoofs.

bnks thinks this approach is worth pursuing. “investigating odd spoof №s ‘d be useful in cogging the structure of odd perfect №s, iffey exist,” he said. “and if odd perfect №s don’t exist, the study of odd spoof №s mite lead to a proof o'their nonexistence.”

other opn experts, including voite and jenkins, are less sanguine. the byu team did “a gr8 job,” voite said, “but i’m not sure we’re any closer to having a line of attack onna opn problem. tis indeed a problem for the ages, [and] perhaps 'twill remain so.”

paul pollack, a mathematician atta university of georgia, is also cautious: “it ‘d be gr8 if we ‘d stare atta list of spoofs n'see some property and somehow prove there are no opns with that property. that ‘d be a presh dream if it works, but it seems too good to be true.”

tis a long shot, nielsen conceded, but if mathematicians are ever goin to solve this ancient problem, they nd'2 try everything. besides, he said, the concerted study of spoofs is just gettin started. his group took some early steps, and they already discovered unexpected properties of these №s. that makes him optimistic bout uncovering even + “hidden structure” within spoofs.

already, nielsen has identified one possible tactic, based onna fact that every spoof found to date, except for descartes’ original ex, has at least one neg base. proving that all other spoofs must ‘ve a neg base ‘d in turn prove that no opns exist — since the bases of opns, by definition, must be both + and prime.

“that sounds like a harder problem to solve,” nielsen said, cause it pertains to a larger, + general category of №s. “but sometimes when you convert a problem to a seemingly + difficult one, you can see a path to a solution.”

patience is required in № theory, where the ?s are often easy to state but difficult to solve. “you ‘ve to think bout the problem, maybe for a long while, and care bout it,” nielsen said. “we're making progress. we’re chipping away atta mountain. na hope s'dat if you keep chipping away, you mite eventually find a diamond.”

original content at: www.quantamagazine.org…
authors: steve nadis

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