`> `
**# Fer_1640.mws **

__A Public Lecture__
**:**
** **
**The history of Fermat [1601-1665] numbers from August 1640**

__Question__
**: **
** What**
** is this lecture **
**about**
**?**

__Answer__
**: It is about one of the most beautiful questions in Mathematics (going back to August 1640),**

** a question that is simple **
**to explain**
**, **
**to state**
**, but for which there is absolutely no sign of an **

** answer. It is about a question which **
**may never be answered**
**. Its answer - if ever there be **

** one - will require creativity of unimaginable proportions.**

Recently - cycling over the
**Luke Kelly Bridge**
** - I came upon someone who said: "John, you have **

** one minute in which to tell me what this Fermat business is all about," and this was my reply: ... ... **

** Now you may - if you wish - go home.**

__Speaker__
**: Dr. John Cosgrave.**

**Address: Mathematics Department, St. Patrick's College, Drumcondra, Dublin 9, IRELAND.**

**e-mail**
**: [email protected] (College) or [email protected] (home)**

**Web**
**: <http://www.spd.dcu.ie/johnbcos>**

__Venue__
**: Room E 201, St. Patrick's College, Drumcondra, Dublin 9, IRELAND.**

__Note__
**. This lecture has been prepared using the Computer Mathematical System **
**Maple**
**, which is researched and developed **

**by Waterloo University (Canada), INRIA (France) and the ETH (Zurich). **
**Maple**
** is used by all academic mathematics students **

**here in St. Patrick's College. The **
**Maple**
** files of this talk **

**in **
__active __
__mws__
** format (which allows a user who has **
**Maple**
** to make his/her own alterations), or **

**in **
__html__
** **
**format (which allows anyone who doesn't have **
**Maple**
** - but who has a web facility - **

to at least read the text of this lecture),

and many other
**Maple**
** worksheets:**

**1st. year **
**Maple**
**/Calculus/Analysis**

**2nd year Number Theory [under construction]**

**3rd year Number Theory and Cryptography**

**some public or seminar/conference lectures**

**together with course notes, examination papers (written and **
**Maple**
** lab), may be accessed/downloaded at my web site.**

**students in St. Patrick's College, Drumcondra. **
**Most**
** of our students are taking the B.Ed. degree, with a view to becoming **

**primary school teachers**
** in Ireland; we also have a number of B. A. students.]**

**I believe it has been remarked (by whom?) that in a once-off lecture what one ought to do it this:**

**1. Tell**
** **
**your listeners what you are going to tell them, **

**which in my case is: I am going to tell you about the historical background to a mathematically significant discovery (made **

**here - on computer #17 in Room D318 of St. Patrick's College, Drumcondra - over the weekend of Frid. 23rd. to Sunday **

**25th. July 1999), namely that the 382447th. **
**Fermat **
**number is **
**composite**
**. **

**That Fermat number - and you will know its meaning shortly - is **
__so utterly vast__
** that were one to write it out in **
**decimal notation**
** **

**(i.e. in the normal, everyday way: ... 79324 ... 872349100218 ... 7777766655443221 ... etc.) at - let us say - 4 digits per inch - and **

**not just on a single sheet of paper, but on sheet upon sheet, stacked back-to-back (at - say - 250 sheets per inch of depth - the **

**thickness of photocopying paper) to form a giant cube, then that cube would have side length roughly 10000000000000....000 **

**(38,358 noughts!!) light years (1 light year - the distance light travels in a year - is approximately 58,656,960,000,000 miles). **

__That is big__
**; the entire universe would fit inside a cube whose side was only (!) about 30,000,000,000 light years. **

**2. Tell them, what you told them, you would tell them: that corresponds to the hyperlinked topics just coming up.**

**3. Tell them, you have told them, what you told them, you would tell them: that will come at the end.**

**1. The term '**
**number**
**' has a variety of meanings. **

__Convention__
**. In what follows (for simplicity, and relevance) whenever I use the term **
**number**
** I will **
__generally mean__
** - unless **

**otherwise stated - what is called a '**
**natural**
** ' number: The **
__natural__
** numbers are the numbers: 1, 2, 3, 4, 5, 6, ... , **
**n**
**, ... **

__Addition:__
** **

`> `
**3 + 8;**

`> `
**9876543213333334444449999999999999999999999999888888899999999998888885555555554444444433333328888899999999999998888888888877777777766666666 + 22222244444444449999999999999923800000999999999999999999999776599999999999999999999999432987;**

__Multiplication__
**. Multiplying 3 and 8 - denoted by 3*8 in **
**Maple**
** - produces 24.**

`> `
**3*8;**

`> `
**98765432133333347777111111111111111111111111111111177777777777777333333333333333334444498888888888888877777777776666655676778876656 ***

23877699999999999999999999999999999999990000000000005432987999999999999988888887765546667745332000000000000;

**2. **
__Powers__
**. By "2 to the power of 5" (say) is meant the **
**repeated**
** product 2*2*2*2*2 (where there are five two's), which is 32. The standard mathematical notation for 2 to the power of 5 is **
**. By "3 to the power of 4" (say) is meant the repeated product 3*3*3*3 (where there are four three's), which is 81. The notation for 3 to the power of 4 is **
**. The general notation for "**
**a**
** to the power of **
**b**
**" - namely **
**a**
*****
**a**
*****
**a**
***...***
**a**
** (where there are **
**b**
** occurences of '**
**a**
**') - is **
**.**

__A special case__
**. In the case where the raising power **
**b**
** is 2, **
** is generally called "**
**a**
** **
__squared__
**."**

__Examples__
**.**

`> `
**2^5; # the first one above**

`> `
**3^4; # the second one above**

`> `
**5^2; # 5 squared:**

`> `
**19999999988888888777776255244423123100000000006666666665555555555555544444444444444444444411111111111111444444444444444012345678987654321^2;**

__Definition of the Fermat numbers__
**. I can now tell you what the **
**Fermat numbers**
** are [in so doing I am "pulling them out of a hat," because I want to get down to basics as quickly as possible; later I will show you how they arose in a most natural fashion] ; the Fermat numbers are the **
**infinite**
** sequence of numbers **
__formed by__
**:**

__starting__
** with the number 2**

__repeatedly form successive squares__
**:**

**,**

**,**

**,**

**,**

**, **

**, etc**

__add 1 to every one of the above numbers__
**, starting with the initial '2,' forming the **
__'__
__Fermat__
__' numbers__
**:**

3, 5, 17, 257, 65537, 4294967296,
**, etc**

The standard mathematical symbols for those numbers are :
**, etc**

**There is a succinct mathematical description for these numbers - based upon elementary observations like:**

** . . . , **
**, **
**, ... , **
** (the completely general case)**

**- namely that the **
**n**
**-th Fermat number (denoted by **
**) has this value:**

** **
** (for **
**, ...)**

`> `
**F[0] := 2^(2^0) + 1; # is the "zero-th" Fermat number, the initial one**

`> `
**F[1] := 2^(2^1) + 1; # is the "first" Fermat number**

`> `
**F[2] := 2^(2^2) + 1; # is the "second" Fermat number**

`> `
**F[3] := 2^(2^3) + 1; # is the "third" Fermat number**

__Just a pause on that last calculation__
**. There I was asking **
**Maple**
** to compute the 3rd. Fermat number, **
**, whose value is given by **
**. **
__Check__
**: **
** is 8, and so **
** is **
**, which is 2*2*2*2*2*2*2*2, which is 256. Thus **
** = 256 + 1 = 257.**

** **

`> `
**F[4] := 2^(2^4) + 1;**

`> `
**F[5] := 2^(2^5) + 1;**

`> `
**F[6] := 2^(2^6) + 1;**

`> `
**F[7] := 2^(2^7) + 1;**

`> `
**F[8] := 2^(2^8) + 1;**

`> `
**F[9] := 2^(2^9) + 1;**

`> `
**F[10] := 2^(2^10) + 1;**

`> `
**F[11] := 2^(2^11) + 1;**

`> `
**F[12] := 2^(2^12) + 1;**

`> `
**length(F[12]); # the command to show F[12] has 1234 digits:**

**The Fermat numbers grow in size extremely rapidly, and it is elementary to argue that each one has**

**either twice as many digits as the previous one**

**or one fewer than twice as many digits as the previous one**

**which I merely illustrate with:**

`> `
**length(2^(2^4) + 1);**

`> `
**length(2^(2^5) + 1);**

`> `
**length(2^(2^6) + 1);**

`> `
**length(2^(2^7) + 1);**

`> `
**length(2^(2^8) + 1);**

`> `
**length(2^(2^12) + 1);**

`> `
**length(2^(2^13) + 1); # 2467 is 2*1234 - 1**

**I hope you can begin to appreciate how it is that the 382447th. Fermat number - **
** - can be so large, can have so many digits: **

**I have done a rough calculation (see the Fermat Number Record section of my web site) which establishes that **
** has approximately **

** digits.**

**So far I have only told you **
__what__
** the Fermat numbers are, and I have yet to tell you **
__how__
** they came about in the first place, and **

**what it is about them that has created such interest in them. That will come shortly. First, though, we need to have a look at:**

__Division__
**. Choose two numbers - **
__95__
** and **
__7__
** (say) - and "**
__divide__
**" the larger ( **
__95__
** ) by the smaller ( **
__7__
** ):**

** **
__95__
** = **
__7__
*****
**13**
** + **
**4**
** ... (i)**

**It is usual to say that "**
__7__
** divides into **
__95__
** thirteen times (the **
**13**
** is the **
__quotient__
**), and leaves **
__remainder__
** 4 over. One then says "7 does **

**not divide evenly**
** into 95" [though mathematicians never say "evenly": you will hear them say: "95 is not divisible by 7." You may **

**well say that mathematicians are odd that they don't say evenly.]**

**Now choose two other numbers - **
__98__
** and **
**7**
** - and divide the larger ( **
__98__
** ) by the smaller ( **
__7__
** ):**

** **
__98__
** = **
__7__
*****
**14 **
**... (ii)**

**It is usual to say that "**
__7__
** divides evenly into **
__98__
**" (**
**14**
** times; the **
**14**
** is the **
__quotient__
**).**

**In case (i) one says there is a **
__remainder of 4__
** when 95 is divided by 7.**

**In case (ii) one says there is a **
__remainder of 0__
** when 98 is divided by 7.**

**Here are the **
**Maple**
** commands for finding those remainders:**

`> `
**95 mod 7;**

`> `
**98 mod 7;**

`> `
**2^5 mod 7;**

`> `
**2^980 mod 816654321900000000055555554444444333333222222211111177777777777777666665555555558765555444491;**

**but look at this (identical to the previous one, except that I have altered the '980' to ... ):**

`> `
**2^98076054328 mod 816654321900000000055555554444444333333222222211111177777777777777666665555555558765555444491;**