10

1

2

5

1 = 10 10 , 2 = 10 5 1 = \large\frac{10}{10}

ormalsize , 2 = \large\frac{10}{5}

ormalsize 1 = 1 0 1 0 ​ , 2 = 5 1 0 ​

5 = 10 2 5 = \large\frac{10}{2}

ormalsize 5 = 2 1 0 ​

10

10

6

28

496

8128

6 = 1 + 2 + 3 ,

28 = 1 + 2 + 4 + 7 + 14 ,

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

300

36

If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

1

2

4

7

( the sum ) × ( the last ) = 7 × 4 = 28 , the sumthe last

1

2

4

8

16

31

31

16

496

1 + 2 + 4 + . . . + 2 k − 1 = 2 k − 1 1 + 2 + 4 + ... + 2^{k-1} = 2^{k} - 1 1 + 2 + 4 + . . . + 2 k − 1 = 2 k − 1 .

If, for some k > 1 , 2 k − 1 k > 1, 2^{k} - 1 k > 1 , 2 k − 1 is prime then 2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k} - 1) 2 k − 1 ( 2 k − 1 ) is a perfect number.

100

(

)

Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.

(

)

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

(

)

... ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands....

a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

(1) The n n n th perfect number has n n n digits.

(2) All perfect numbers are even.

(3) All perfect numbers end in 6 and 8 alternately.

(4) 2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k} - 1) 2 k − 1 ( 2 k − 1 ) , for some k > 1 k > 1 k > 1 , where 2 k − 1 2^{k} - 1 2 k − 1 is prime.

(5) There are infinitely many perfect numbers. Theth perfect number hasdigits.All perfect numbers are even.All perfect numbers end inandalternately. Euclid 's algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form, for some, whereis prime.There are infinitely many perfect numbers.

(1)

(3)

(2)

(4)

(5)

(4)

(

)

There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 , 1024 , 2048 , 4096 ; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity.

6

28

496

8128

... only one is found among the units, 6 , only one other among the tens, 28 , and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128 . And it is their accompanying characteristic to end alternately in 6 or 8 , and always to be even.



When these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next. ... the result is 496 , in the hundreds; and then comes 8128 in the thousands, and so on, as far as it is convenient for one to follow.

6

28

(354

430)

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect ...

2 n p 2^{n}p 2 n p

p p p

2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k} - 1) 2 k − 1 ( 2 k − 1 )

2 k − 1 2^{k} - 1 2 k − 1

(1194

1239)

1500

2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k} - 1) 2 k − 1 ( 2 k − 1 )

k k k

1509

2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k} - 1) 2 k − 1 ( 2 k − 1 )

k k k

(

)

1461

1461

1458

1460

1536

Ⓣ ( Both kinds of arithmetic )

2 11 − 1 = 2047 = 23.89 2^{11} - 1 = 2047 = 23 . 89 2 1 1 − 1 = 2 0 4 7 = 2 3 . 8 9

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

2 13 − 1 = 8191 2^{13} - 1 = 8191 2 1 3 − 1 = 8 1 9 1

(

)

2 12 ( 2 13 − 1 ) = 33550336 2^{12}(2^{13} - 1) = 33550336 2 1 2 ( 2 1 3 − 1 ) = 3 3 5 5 0 3 3 6

8

6

8

1555

1977

1603

800

750

(

132

)

2 17 − 1 = 131071 2^{17}- 1 = 131071 2 1 7 − 1 = 1 3 1 0 7 1

(

75 0 2 = 562500 > 131071 750^{2} = 562500 > 131071 7 5 0 2 = 5 6 2 5 0 0 > 1 3 1 0 7 1

131071

)

2 16 ( 2 17 − 1 ) = 8589869056 2^{16}(2^{17} - 1) = 8589869056 2 1 6 ( 2 1 7 − 1 ) = 8 5 8 9 8 6 9 0 5 6

6

8

6

2 19 − 1 = 524287 2^{19} - 1 = 524287 2 1 9 − 1 = 5 2 4 2 8 7

(

75 0 2 = 562500 > 524287 750^{2} = 562500 > 524287 7 5 0 2 = 5 6 2 5 0 0 > 5 2 4 2 8 7

)

2 18 ( 2 19 − 1 ) = 137438691328 2^{18}(2^{19} - 1) = 137438691328 2 1 8 ( 2 1 9 − 1 ) = 1 3 7 4 3 8 6 9 1 3 2 8

Ⓣ ( Both kinds of arithmetic )

p p p

2

3

5

7

13

17

19

23

29

31

37

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

p p p

2

3

5

7

13

17

19

23

29

31

37

1638

... I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3 , 7 , 11 , 13 , one would have 198585576189 , which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers...

1636

1640

... here are three propositions I have discovered, upon which I hope to erect a great structure. The numbers less by one than the double progression, like

1 2 3 4 5 6 7 8 9 10 11 12 13

1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 let them be called the radicals of perfect numbers, since whenever they are prime, they produce them. Put above these numbers in natural progression 1 , 2 , 3 , 4 , 5 , etc., which are called their exponents. This done, I say When the exponent of a radical number is composite, its radical is also composite. Just as 6 , the exponent of 63 , is composite, I say that 63 will be composite. When the exponent is a prime number, I say that its radical less one is divisible by twice the exponent. Just as 7 , the exponent of 127 , is prime, I say that 126 is a multiple of 14 . When the exponent is a prime number, I say that its radical cannot be divisible by any other prime except those that are greater by one than a multiple of double the exponent... Here are three beautiful propositions which I have found and proved without difficulty, I shall call them the foundations of the invention of perfect numbers. I don't doubt that Frenicle de Bessy got there earlier, but I have only begun and without doubt these propositions will pass as very lovely in the minds of those who have not become sufficiently hypocritical of these matters, and I would be very happy to have the opinion of M Roberval.

18

1640

p p p

a a a

p p p

a p − 1 − 1 a^{p-1}- 1 a p − 1 − 1

p p p

1640

2 23 − 1 2^{23} - 1 2 2 3 − 1

(

2 23 − 1 = 47 × 178481 2^{23} - 1 = 47 \times 178481 2 2 3 − 1 = 4 7 × 1 7 8 4 8 1

)

2 37 − 1 2^{37} - 1 2 3 7 − 1

(

2 37 − 1 = 223 × 616318177 2^{37} - 1 = 223 \times 616318177 2 3 7 − 1 = 2 2 3 × 6 1 6 3 1 8 1 7 7

)

(

)

1 0 20 10^{20} 1 0 2 0

1 0 22 10^{22} 1 0 2 2

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

p p p

2 37 − 1 2^{37} - 1 2 3 7 − 1

2 37 − 1 2^{37} - 1 2 3 7 − 1

1640

( i ) If n n n is composite, then 2 n − 1 2^{n} - 1 2 n − 1 is composite.

( ii ) If n n n is prime, then 2 n − 2 2^{n} - 2 2 n − 2 is a multiple of 2 n 2n 2 n .

( iii ) If n n n is prime, p p p a prime divisor of 2 n − 1 2^{n}- 1 2 n − 1 , then p − 1 p - 1 p − 1 is a multiple of n n n .

(

)

(

)

(

)

p p p

2

37

1

37

p − 1 p - 1 p − 1

p p p

2 × 37 m + 1 2 \times 37m+1 2 × 3 7 m + 1

m m m

p p p

149

(

)

223

(

m = 3 m = 3 m = 3

)

2 37 − 1 = 223 × 616318177 2^{37} - 1 = 223 \times 616318177 2 3 7 − 1 = 2 2 3 × 6 1 6 3 1 8 1 7 7

1644

Ⓣ ( Thoughts on physical mathematics )

2 p − 1 2^{p} - 1 2 p − 1

(

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

)

p = 2 , 3 , 5 , 7 , 13 , 17 , 19 , 31 , 67 , 127 , 257 p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 p = 2 , 3 , 5 , 7 , 1 3 , 1 7 , 1 9 , 3 1 , 6 7 , 1 2 7 , 2 5 7

p p p

257

... to tell if a given number of 15 or 20 digits is prime, or not, all time would not suffice for the test.

47

p p p

19

258

2 p − 1 2^{p} - 1 2 p − 1

42

5

2 p − 1 2^{p}- 1 2 p − 1

1732

2 30 ( 2 31 − 1 ) = 2305843008139952128 2^{30}(2^{31} - 1) = 2305843008139952128 2 3 0 ( 2 3 1 − 1 ) = 2 3 0 5 8 4 3 0 0 8 1 3 9 9 5 2 1 2 8

125

1738

2 29 − 1 2^{29} - 1 2 2 9 − 1

(

)

(

)

p = 31 p = 31 p = 3 1

p = 29 p = 29 p = 2 9

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

6

8

(

)

1638

( 4 n + 1 ) 4 k + 1 b 2 (4n+1)^{4k+1} b^{2} ( 4 n + 1 ) 4 k + 1 b 2

4 n + 1 4n+1 4 n + 1

2 p − 1 ( 2 p − 1 ) 2^{p-1}(2^{p} - 1) 2 p − 1 ( 2 p − 1 )

p = 41 p = 41 p = 4 1

p = 47 p = 47 p = 4 7

1753

Ⓣ ( Thoughts on physical mathematics )

2 30 ( 2 31 − 1 ) 2^{30}(2^{31} - 1) 2 3 0 ( 2 3 1 − 1 )

150

1811

2 30 ( 2 31 − 1 ) 2^{30}(2^{31} - 1) 2 3 0 ( 2 3 1 − 1 )

... is the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will ever attempt to find one beyond it.

1876

2 67 − 1 2^{67} - 1 2 6 7 − 1

2 127 − 1 2^{127} - 1 2 1 2 7 − 1

2 126 ( 2 127 − 1 ) 2^{126}(2^{127}- 1) 2 1 2 6 ( 2 1 2 7 − 1 )

1930

p = 127 p = 127 p = 1 2 7

2 p − 1 2^{p} - 1 2 p − 1

m = 2 p − 1 m = 2^{p} - 1 m = 2 p − 1

2 m − 1 2^{m} - 1 2 m − 1

2 p − 1 2^{p} - 1 2 p − 1

p = p = p = 3 , 7 , 127 , 170141183460469231731687303715884105727 , ...

(

)

2 p − 1 2^{p} - 1 2 p − 1

p = p = p =

170141183460469231731687303715884105727

1883

2 60 ( 2 61 − 1 ) 2^{60}(2^{61}- 1) 2 6 0 ( 2 6 1 − 1 )

67

61

1903

2 67 − 1 2^{67} - 1 2 6 7 − 1

1903

2 67 − 1 = 147573952589676412927 2^{67} - 1 = 147573952589676412927 2 6 7 − 1 = 1 4 7 5 7 3 9 5 2 5 8 9 6 7 6 4 1 2 9 2 7 .

761838257287

193707721

147573952589676412927

[

(

)

2 67 − 1 2^{67} - 1 2 6 7 − 1

]

1911

2 88 ( 2 89 − 1 ) 2^{88}(2^{89} - 1) 2 8 8 ( 2 8 9 − 1 )

2 107 − 1 2^{107}- 1 2 1 0 7 − 1

2 106 ( 2 107 − 1 ) 2^{106}(2^{107}- 1) 2 1 0 6 ( 2 1 0 7 − 1 )

1922

257

2 257 − 1 2^{257}- 1 2 2 5 7 − 1

(

)

... the existence of [ an odd perfect number ] - its escape, so to say, from the complex web of conditions which hem it in on all sides - would be little short of a miracle.

1888

4

29

300

1 0 6 10^{6} 1 0 6

(2020)

51

2 88 ( 2 89 − 1 ) 2^{88}(2^{89}- 1) 2 8 8 ( 2 8 9 − 1 )

1911

(

)

2 82 589 933 − 1 2^{82 589 933} - 1 2 8 2 5 8 9 9 3 3 − 1

(

)

2 82 589 932 ( 2 82 589 933 − 1 ) 2^{82 589 932} (2^{82 589 933} - 1) 2 8 2 5 8 9 9 3 2 ( 2 8 2 5 8 9 9 3 3 − 1 )

2018

51

23

860

51

51