the sunday tImes of malta maRCh 11, 2018 | 51

LIFE&WELLBEING SCIENCE

ALEXANDER FARRUGIA

You are given a number n, and youare asked to find the two wholenumbers a and b, both greaterthan 1, such that a times b is equalto n. If I give you the number 21,say, then you would tell me “thatis 7 times 3”. Or, if I give you 55,you would say “that is 11 times 5”.

Now suppose I ask you for thetwo whole numbers, both greaterthan 1, whose product (multipli-cation) equals the number6,436,609. This is a much harderproblem, right? I invite you to tryto work it out, before you read on.

A prime number is a numberthat is only divisible by 1 and byitself. The number 21 is notprime, because it is divisible by3 and by 7. However, the num-bers 3 and 7 themselves are bothprime. On the other hand, thenumbers 21, 55 and 6,436,609have this thing in common: eachof them is equal to the product oftwo prime numbers.

The problem of splitting up anumber into its constituentprimes is called the prime factor-ization problem. It is known thatif these prime numbers are largeenough, then there is no efficientway of finding them in a feasible

amount of time. This fact is thebasis of internet security schemessuch as the RSA algorithm used toencrypt online bank transactionsand social network websites suchas Twitter.

Even though the prime fac-torisation problem for the num-ber 6,436,609 looks difficult, itis a relatively straightforwardproblem for a computer. Indeed,in 2009, a 512-bit number was

factored into its two primes byBenjamin Moody – that is anumber having 155 digits! (Bycomparison, the number6,436,609 has only seven digits.)It took his desktop computer 73days to achieve this feat. Later,during the same year, a 768-bitnumber, having 232 digits, wasfactored into its constituentprimes by a team of 13 academ-ics – it took them two years to doso! This is the largest numberthat has been factored to date.

Nowadays, most websitesemploying the RSA algorithmuse at least a 2,048-bit (617-digit) number to encrypt data. Some websites such as Facebook and GMail use a differentencryption algorithm called ECC(elliptic curve cryptography),which is not based on the primefactorisation problem.

By the way, what are the primefactors of 6,436,609? I will notspoil you with the answer, but Iwill give you a hint: one of itsprime factors has three digits.

Alexander Farrugia is a lecturer atthe University of Malta Junior Col-lege with a PhD in mathematics anda top writer on the websitewww.quora.com, where he writesprimarily about various aspects ofmathematics.

DID YOU KNOW?

• G. H. Hardy, in his 1940 book A Mathe-matician’s Apology, wrote: “No one hasyet discovered any purpose to beserved by the theory of numbers andit seems unlikely that anyone will do sofor many years.” Sixty years later, thetheory of numbers is at the heart ofinternet security! Read the main arti-cle for more details.

• The sieve of Eratosthenes is an oldmethod used to generate all the primenumbers up to some number N. Essen-tially, a list of all the numbers from 2to N is produced, then the first numberin this list is placed among the primenumbers and all the multiples of thisnumber are crossed out from the list.This process is repeated until the listis exhausted.

• The number 11 is the smallest so-calledrepunit prime, whose decimal repre-sentation 11 is a string of ones. Thenext decimal repunit prime is1,111,111,111,111,111,111, the number rep-resented by a string of 19 ones. Thismeans, for example, that the number11,111 is not prime. What are the twofactors of 11,111?

For more trivia, see: www.um.edu.mt/think.

MYTHDEBUNKED

Is 1 a primenumber?If this question was askedbefore the start of the 20thcentury, one would haveinvariably received a ‘yes’.Most mathematicians beforethe 20th century listed 1 asone of the prime numbers.However, at the start of the20th century, 1 was debarredfrom being a prime number.Instead, it was placed in itsown category, that of a unit.Why did this change occur?

To answer this, we need tofirst understand what a unitis. A unit is a whole numberthat has a ‘partner’, also awhole number, such thatwhen we multiply the unit byits partner, the answer is 1.From this, we can confirmthat 1 is a unit, having the part-ner 1; indeed, 1x1=1. Moreover,no other positive whole num-ber is a unit, because we can-not multiply any whole num-ber greater than one by anywhole number ‘partner’ andget 1.

The whole numbers satisfya very important propertycalled the Fundamental Theorem of Arithmetic. Thisproperty states that any posi-tive whole number that is nota unit is either a prime num-ber or made up of the multi-plication of some unique listof prime numbers. For exam-ple, the number 59 is a primenumber, but the number 60 ismade up of the primes 2, 2, 3and 5 such that 2x2x3x5=60.In fact, not only is 60 equal to2x2x3x5, but no other list ofprime numbers, when multi-plied by each other, can resultin 60.

Why do mathematiciansexclude 1 from being a primenumber? If 1 were prime,then the number 60 wouldalso be equal to the multipli-cation 1x2x2x3x5 of theprimes 1, 2, 2, 3 and 5. Thismeans that our list of primenumbers whose multiplica-tion is equal to 60 would notbe unique, and this wouldcontradict the Fundamental Theorem of Arithmetic.

PHOTO OF THE WEEK

Prime numbers and cryptography

The 2,160-bit public key, in hexadecimal, of a local website that employsthe RSA algorithm to encrypt its data. As its name suggests, this numberis public information and is downloaded by your web browser and storedon your device as soon as you visit the website.

SOUND BITES

• The day after Christmas of 2017, a 51-yearold electrical engineer from Germantown,Tennessee discovered what, up to this day,is the largest known prime number.Jonathan Pace has been a volunteer of theGIMPS project (Great Internet MersennePrime Search) for 14 years, using free soft-ware available by GIMPS to search for so-called Mersenne primes. A Mersenneprime is a prime number that is equal toone less than a power of two. For example,3 and 7 are Mersenne primes, because theyare equal to 2x2-1 and 2x2x2-1, respec-

tively. Mr Pace’s Mersenne prime is foundby multiplying 77,232,917 twos, and thensubtracting 1! This number, if writtendown, would have a whopping 23,249,425digits! That is almost a million digits morethan the next largest discovered prime,which was discovered in January 2016,also by GIMPS.

• A prime gap is the difference between twosuccessive prime numbers. For example,the numbers 317 and 331 are both prime,but no number in between is prime, so wehave a prime gap of 14. By the Prime Num-ber Theorem, the average prime gapbetween any two consecutive prime num-bers among the first n whole numbers is

the natural logarithm of n. The merit of aprime gap is thus defined as the prime gapdivided by this average prime gap. Forexample, the merit of the prime gapbetween 317 and 331 is 2.43, which meansthat this prime gap is more than twice theaverage prime gap among those between1 and 317. In December 2017, the GapCoinnetwork discovered a prime gap of length8350 following an 87-digit prime. Thisprime gap has merit 41.94, which meansthat it is almost 42 times as large as theaverage prime gap. This is the largestprime gap merit discovered to date.

For more science news, listen to Radio Mochaon Radju Malta 2 every Saturday at 11.05am.

The first 50 prime numbers.

The front page of the very first printedversion of the mathematical treatiseElements. The Elements is a 13-volumebook entirely written by Euclid ofAlexandria (circa 300BC). This treatise isconsidered to contain the very first studyof prime numbers, which Euclid defines inBook 7, Definition 11 as ‘those which aremeasured by a unit alone’. He also provesimportant results on prime numbers, suchas his famous theorem in Book 9,Proposition 20, asserting that ‘primenumbers are more than any assignedmultitude of prime numbers’ – in otherwords, there are infinitely many primenumbers. The version of the Elements inthe photo is in both Greek and Latin andwas published in 1573 in Paris. It iscurrently preserved in the Archives andSpecial Collections of Dickinson College,Carlisle, Pennsylvania. An online versionof an English translation of the entireElements is available for free athttps://mathcs.clarku.edu/~djoyce/java/elements/elements.html and ismaintained by Prof. David Joyce of ClarkUniversity, Worcester. PHOTO:https://www.maa.org/sites/default/files/images/upload_library/46/Swetz_2012_Math_Treasures/Dickinson/EuclidisElementorum_title.png