Prime Numbers Notes

Notes on Prime Numbers

Factors of an integer

By definition, an integer b is a factor of  an integer a, different from zero, iff there exists an integer q such that a = bq.

For instance, 17 is a factor of 102; in this case, q = 6.

Using an appropriate computer program, it can be obtained all of the factors of a given integer.

Examples: the factors of 384 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384. The factors of 497 are 1, 7, 71, 497. The factors of 499 are 1, 499.

The restriction about a in above definition is convenient to prevent problems with the theory of quotient between integers. For the same reason zero cannot be a factor of any integer.

From above definition it follows that if integer b is a factor of  a, then b is a factor of -a, also -b is a factor of a and -b is a factor of -a.

Example: 96 is a factor of 384 (96 × 4 = 384), 96 is a factor of -384 (96 × (-4) = -384), -96 is a factor of 384 (-96 × (-4) = 384) and -96 is a factor of  -384 ((-96) × 4 = -384).

For any two positive integers a, b such that b ≤ a it can be done a list of  multiples of b not greater than a. If bq is the greatest in that list then r = a – bq, the reminder, is a nonegative integer and r < b.

Example: a = 38, b = 9. Multiples of 9 not grater than 38 = {9, 18, 27, 36}. The greatest of them is 36 = 9 × 4. In this case,  r = 38 – (9 × 4) = 2.

The following schemas show a way to present graphically the relation between numbers a, b, q, r  in  r = a – bq

Example:

or

Example

This situation can be extended for all of the integers, as in the following examples:

a = 38, b = -9. In this case, 2 = 38 – (-9) × (-4)
a = -38, b = -9. In this case, -2 = -38 – (-9) × 4
a = -38, b = 9. In this case, -2 = -38 – 9 × (-4)

From now on, only positive integers will be considered.

A property to look at

If r = 0 in  r = a – bq, it means that b is a factor of a. If a = bq then a + 1 = bq + 1 

Then, 1 = (a + 1) – bq. This means that 1 is the remainder for a + 1 divided by b. 

A conclusion from above considerations is: if  a = bq then b is not a factor of  a + 1.

If b is a factor of  a then b is not a factor of  a + 1.

Order and factors

If b is a factor of a then b ≤ a. The converse, of course, is not true.

Effectively,  b ≤ a means that b + h = a for a no negative integer h. If  b is a factor of a then bq = a where q is a positive integer. 

If  b = a also b ≤ a; in this case, q = 1, h = 0.

If b ≠ a, then bq = a can be written as b + b(q -1) = a which means that b ≤ a. 
In this case, h = b(q -1), where q > 1.

Example: 71 is a factor of  497 (71 × 7 = 497) and 71 + 71 × (7 – 1) = 497. Therefore 71 ≤ 497.

Concept of prime number

In above examples about the factors of integers it can be seen that the list of factors of a number includes 1 and the number itself. 

This happen for any positive integer n: 1 is a factor of n since 1 × n = n. Also n is a factor of n since n × 1 = n. 

1 and n are called trivial factors of a positive integer n. 

A  number  p is prime iff their factors are 1 and p only.

In above examples, 499 is prime.

The sieve of Eratostenes

A method to obtain a list of succesive prime numbers consists in writing a list of positive integers from 2 on, as large as desired. Next, croos out all the multiples of 2, all the multiples of 3, all the multiples of 5, and so on, continuing with the next number not crossed. The remaining numbers constitute a list of primes. 

The following is a sieve for integers between 2 and 30. The excluded numbers are in color, different of black.

2, 3, 4, 5, 6, 7, 8, 9  10 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

There are in red the multiples of 2; in blue, the multiples of 3 which are not multiples of 2; in purple, the multiples of 5 which are not multiples of 2 or 3.In black, the prime numbers, from 2 not greater than 30: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}.

Frequencies and histograms

The set of prime numbers not greater than 100 is
{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}.

For such a set of data it can be done a frequency table like the following:

Interval	Frecuency
1 a 20	8
20 a 40	4
40 a 60	5
60-80	5
80-100	3

And the information can be presented as an histogram like the following:

In the following list (obtained from Wolfram Mathematica) are included the first 420primes between 1 and 3000.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897

From this list it can be registered the frequency with which prime numbers appear in successive intervals of 100 integers as it can be seen in the following table.

Interval	Frecuency
1 a 100	25
100 a 200	21
200 a 300	16
300 a 400	16
400 a 500	17
500 a 600	14
600 a 700	16
700 a 800	14
800 a 900	15
900 a 1000	14
1000 a 1100	16
1100 a 1200	12
1200 a 1300	15
1300 a 1400	11
1400 a 1500	17
1500 a 1600	12
1600 a 1700	15
1700 a 1800	12
1800 a 1900	12
1900 a 2000	13
2000 a 2100	14
2100 a 2200	10
2200 a 2300	15
2300 a 2400	15
2400 a 2500	10
2500 a 2600	11
2600 a 2700	15
2700 a 2800	14
2800 a 2900	12

A corresponding hystogram is the following:

It can be seen certain regularity if one observe them with an elastic criterion not enough however to predict a general behavior.

It can be perceived nevertheless a light tendency to decrease in the number of prime items as the sequence advance.

Distance between two consecutive prime numbers

Experimentally it can be studied another chatacteristic, consisting in determine the differences between  each two  consecutive prime numbers from a given list. For prime numbers not grater than 100 a greatest distance of 8 places is obtained. It is produced between 89 and 97. It follows a list of named distances. 

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8

For the first 420 prime numbers the corresponding distances are:

1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,6,4,2,10,14,4,2,4,14,6,10,2,4,6,8,6,6,4,6,8,4,8,10,2,10,2,6,4,6,8,4,2,4,12,8,4,8,4,6,12,2,18,6,10,6,6,2,6,10,6,6,2,6,6,4,2,12,10,2,4,6,6,2,12,4,6,8,10,8,10,8,6,6,4,8,6,4,8,4,14,10,12,2,10,2,4,2,10,14,4,2,4,14,4,2,4,20,4,8,10,8,4,6,6,14,4,6,6,8,6,12,4,6,2,10,2,6,10,2,10,2,6,18,4,2,4,6,6,8,6,6,22,2,10,8,10,6,6,8,12,4,6,6,2,6,12,10,18,2,4,6,2,6,4,2,4,12,2,6,34,6,6,8,18,10,14,4,2,4,6,8,4,2,6,12,10,2,4,2,4,6,12,12,8,12,6,4,6,8,4,8,4,14,4,6,2,4,6,2,6,10,20,6,4,2,24,4,2,10,12,2,10,8,6,6,6,18,6,4,2,12,10,12,8,16,14,6,4,2,4,2,10,12,6,6,18,2,16,2,22,6,8,6,4,2,4,8,6,10,2,10,14,10,6,12,2,4,2,10,12,2,16,2,6,4,2,10,8,18,24,4,6,8,16,2,4,8,16,2,4,8,6,6,4,12,2,22,6,2,6,4,6,14,6,4,2,6,4,6,12,6,6,14,4,6,12,8,6,4,26,18,10,8,4,6,2,6,22,12,2,16,8,4,12,14,10,2,4,8,6,6,4,2,4,6,8,4,2,6,10,2,10,8,4,14,10,12,2,6,4,2,16,14,4,6,8,6,4,18,8,10

A corresponding histogram is the following:

In above list it can be observed notable variations in studied distances. For instance, the distances not greater than 5 are the ones with more frequency, followed by the ones between 5 and 10, between 10 and 15,and so on.

The named distance is 1, only for the first two numbers in the list, because in the  remaining cases if 1 is added an even number is obtained. The number 2 is the only even number which is prime.

The distances, or number of places separating a prime to the next prime in the above sequence, varie between 1 and 34. The distance 34 corresponds to consecutive prime numbers 1327 and 1361, printed in red in above sequence.

These investigations, with statistic flavor, have the aim of finding some regularity which allows to predict the behavior of the sequence of prime numbers and, may be, to find a formula to generate prime numbers or determine how many primes are there which are less than any given integer.

Finite or infinite number of primes?

In “Introduction to Number Theory”, T. Nagell states that there are not primes between 1327 and 1361. A jump of 34 places, as we have seen before. Also he finds the same jump for 8467 and 8501.

Starting with the above list of 420 primes, it cannot be predicted a lot about the behavior of this sequence, except that it is increasing and it seems that the number of items decrease as the numbers in the  natural sequence increase. In lists that have been obtained, with or without computers, it can be seen the tendency to scarcity for greater values. One of those lists, with 10.000 items was done in 1657. Another, in 1770 had 102.000 elements (Taken from “Recreations in the theory of Numbers” by A. H. Beiler.) .

Powerful computers have produced millions of prime numbers. Nevertheless the named scarcity persists. For instance, there are not prime numbers between 396733 and 396833, a 100 places jump. This could induce to think that for a sufficiently great number there are not primes at all.

 However, a theorem included in Euclid's Elements, book IX, states that the number of primes is infinite.

To be proved: given any prime number p, it is always feasible  to find at least one prime number greater than p.

The proof  is by contradiction.

The prime numbers can be listed as follows: p₁ = 2, p₂ = 3, p₃ = 5, …, p_t .

Let p_t be the gratest prime number.

Construct the number m such that p₁ × p₂ × p₃ × … × p_t = m.

That is, 2 × 3 × 5 × … × p_t = m.

It follows that every prime in the list p₁ , p₂ , p₃ , … , p_t  is a factor of m. But no item of that list can be a factor of m + 1 (remainder would be 1).

If it happens that m + 1 is prime, then  at least a prime greater than p_t (the supposed greatest prime in the list) have been found.

If it happens that m + 1 is not prime then it must exist a prime factor p_v of  m + 1, different from each of  p₁, p₂, p₃, …, p_t which necessarily had to be greater than p_t  (the supposed greatest prime in the list).

In each of above situations a contradiction with the fact that p_t is the greatest prime number in the list is obtained.

The conclusion is that the number of primes is infinite.

 For a better undersatanding of above reasoning, suppose that p_t = 7 is the greatest of all the primes. The product 2 × 3 × 5 × 7 (i.e. 210) has 2, 3, 5, 7 as factors. None of them is factor of 211. As 211 resulted to be prime, there is a situation in which has been found a prime greater than the supposed gratest prime, in this case 7.

For p_t = 13 as the greatest of all the primes, the corresponding product is 
2 × 3 × 5 × 7 × 11  × 13 (i.e.  m =30030) . The number 3031 is not prime; its factors are 1, 59, 509, 30031. The number 59 is prime and is grater than 13. 

Formulae to obtain prime numbers

There have been many intends to find  formulas to generate prime numbers some of them are useful to certain extent. One of them is n² + n + 17  which produces primes for values uf n  from 1 to 16: the primes obtained are, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289. For n = 17 the formula gives 323 which is not prime; their factors are 1, 17, 19, 323. But for n = 18 the formula gives 359, which is prime. Also for n = 50 it gives 2657, which is prime. For n = 25 the formula gives 667 which is not prime; its factors are 1, 23, 29, 667. The formula is unsure.

Euler's formula n² + n + 41 gives prime numbers for n not greater than 40:

43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503,

547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523,

1601. 

For n = 41 the formula gives 1763, which is not prime; its factors are 1, 41, 43, 1763.

The function π(n)

For a positive integer n this function gives the number of primes not greater than n. The following table gives some values of π(n) for certain values of n. They can be obtained directly from the above list of 420 prime numbers.

n	10	100	1000	2000	2900
π(n)	4	25	168	303	420

The use of the grrek character π is traditional. It has not relation with the numeric value associated with a circle for instance.

Legendre stated a good approximation: π(n) ~ n/ln n

The following table gives rational approximations for considered instances. L is useful only for large values of n.

n	10	100	1000	2000	2900
π(n)	4	25	168	303	420
L	4.343	21.715	144.765	263.127	363.72

Gauss used a more precise approximation with a formula involving a logarithmic integral

Modern technology faccilitates to obtain precise and fast results like  the former or determine if a number is prime and identify the next prime in a given list.

The fundamental theorem of arithmetic

This theorem states that each integer grater than 1 either is prime or composite. In the last case it can be expressed uniquely as the product of their prime factors. 

In the named product a factor can appear one or more times. The uniqueness of factoring does not refer to the order in which factors appear. This means that each integer is completely characterized by the set of its prime factors.

For instance, {2, 3, 2, 7, 5} characterizes the integer 420. Te set  {5, 7, 7, 11, 5, 2, 7} characterizes the integer 188650.

A question to be asked could be: are there any rare number which is not prime and cannot be expresed as a product of primes?

The answer is no, and this can be sustained by contradiction.

Suppose that m is the least of such rare numbers. Not being prime, it has at least two factors a, b, other than 1. The numbers a, b are smaller than m; so, they are not rare. If both of a, b are not primes, then at least one of them can be factored in primes. This means tha m is not rare. With this contradiction it has been proved that rare numbers do not exist.

Symbolically, if n is an integer greater than 1, then n can be expressed with the next product formula:

Where p₁, p₂, …, p_r are the prime factors of n and α_i are natural numbers which indicate the number of times of appearing of each p_i.

For example, 188650 = 2¹ × 5² × 7³ × 11¹.

Conjectures

Some important conjectures aboutprime numbers have been formulated:

Every even integer n, greater of equal to 6 can be expressed as the sum of  two odd primes

Example: 36 = 7 + 29 (also 13 + 23, 17 + 19)

Every odd number n greater or equal to 9 can be expressed as the sum of three odd primes.

Example: 35 = 3 + 13 + 19 (also 7 + 11 + 17)

There are many pair of primes which differ in two units. For instance, (5, 7), (29, 31), (1019,1021), (2591, 2593). They are called twin primes. A

Another conjecture is: there are infinite twin primes.

A final conjecture for now: there are infinite palindrome primes. Examples: 11, 101, 131, 797, 3113.

Prime numbers and numeric congruences

Fermat stated the following theorem: if p is a prime number, then x^p is congruent to x modulo p, for any positive integer x. 

This means that p is a factor of  x^p – x.

For instance, 3 is a factor of 8³ – 8. (3 is a factor of 504). In other words, 8³ is congruent to 8, modulo 3.

Congruence modulo 3 allows to classify the set of positive integers in three classes, namely,

class of 0 = {0, 3, 6, 9, 12, 15,…}

class of 1 = {1, 4, 7, 10, 13, …}

class of 2 = {2, 5, 8, 11, 14, …}

To know in which of these classes is,for instance, the number 7435, it is enough to divide 7435 by 3 and obtain the reminder; in this case, the remainter is 1. So, 7435 is congruent with 1, modulo3.

Para saber a cuál de estas clases pertenece, por ejemplo, el número 7435, basta dividir a este número por 3 y observar cuál es el residuo. En este caso, el residuo es 1. Por lo tanto, se puede asegurar que 7435 está en la clase del 1. Dicho de otro modo, 7435 es congruente con 1, modulo 3.

Another example: 5 must be a factor of  12⁵ - 12. (5is a factor of 248820) This means that 12⁵is congruent with 12, modulo 5.

Congruence modulo 5 allows to classify the set of positive integers in five classes. To know in which of these classes is, for instance, the number 7435, it is enough to divide by 5 and obtain the reminder. In this case is 0: {0, 5, 10, 15, ...}.

A secret code

A way to make a secret code is to assign a numeric code without giving the (prime) number which acts as modulo. For instance,  the number  43617 is in the class 0 (or 3), if the modulo is 3. But is in the class 2, if the modulo is 5.

The modulo may be a number with many figures which might turn difficult to guess the class to which a given number belongs. For instance, if the number acting like modulo is 1033 then the number 487938 is in the class of 362, one of the 1033 classes of the partition.

Modern strategies are more sophisticate, but they are relation with numeric properties as the former.

Merssenne and Fermat  numbers

A Mersenne number is a prime of  the form 2^p – 1, where p is a prime. With values 2, 3, 5, 7, can be obtained Mersenne's primes. For p = 11 the formula gives 2047 which is not prime (23 and 89 are factors of it). For p = 13 the number obtained is 8191, which is prime.

Another conjecture is: there are infinite Mersenne's primes.

If  2ⁿ + 1 is a prime number, then n is a power of 2 and the number obtained is a  Fermat's prime number. examples: for n = 4 = 2² the result is 17. For n = 8 = 2³ the result is 257. For n = 16 = 2⁴ the result is 257, which is prime. For n = 16 = 2⁴ the result is 131073, not prime (3 and 43691 are factors of it).

Peano's Formulario

An item of Peano's Formulario

Introduction

Giusseppe Peano was born in 1858 in Spinetta,  Cuneo province,  Italian NE,  near France. Peano attended  Turín University and later was a mathematics instructor there and in the military Academy.
Peano's  mathematics contributions in logic, geometry, differential and integral calculus are included in several articles and  books. In their publications he makes mention of Cantor, Fermat, Leibnitz, Frege and other matematicians. From Frege he adopts several symbols. Peano was admired by contemporary colleagues like Bertrand Russell. Peano lived all of his life in Turin where  he was a student, a teacher,  a family former, and a school creator. He died at the age of 74.

The "Formulario" project

The "Formulario" was first published in french, in 1894: "Formulaire de mathématiques". There were several successive versions between 1894 and 1900. The item referred in this article has not date or place of publication. It only states, that the content is I and II instalments.

The referred publication is inscribed in an ambitious project of mathematical encyclopedia. The language used is "latino sine flexione", a simplified version of latin invented by Peano with the intention of creating a universal language like "interlingua", a parallel intend done in several countries at that time. Peano invented also a set of mathematics symbols, some of them were adapted by Russell and others. There are a number of such symbols in modern mathematics books.

The next illustration shows part of one of the initial pages of the referred item.
   

At the end of chapter on logic, "Vocabulario I", there are abbreviated references of the used languages: English (A, for anglo), German (D for Deutsh or G for Germano), French (F for Franco), Spanish (H for Hispano), Italian (I for Italo), Russian (R for Russo) , Latin (L for Latino). G could indicate Greek, and S for Sanskrit. It can be seen the author's intention for inclusion of all of the mentioned languages in relation with their writings. Symbolic statements are also accompanied with the described references.

The chapters included in the referred item are: Mathematical Logic, Arithmetic, Algebra, Geometry, Limits, Differential Calculus, Integral Calculus, Curve Theory. There are few illustrations; text and specialized symbolic language predominate.

It follows a brief description of each of the mentioned chapters.

Mathematical Logic (chapter I in Formulario)

The first of seven parts of this chapter deals with equal relation and logic connectives, now expressed with "then" and "and". Dots indicate separation between parts of formulae, following Leibnitz. Each symbolic statement is accompanied with examples or explanations in "latino sine flexione", naturally. As an example is transcribed the following:

Nos pote scribe  primo propositione symbolico:

.1            x=x

Lege:     “x aequa x”.       Numero .1 es numero de propositione

Comments, examples or explanations pretend to reinforce undefined concepts or statements taken as axioms for the theory.

Next, another illustration taken from the item.

Part two is about the concept of class (Cls) and the belonging relation of an object with respect to a class. A greek epsilon is the corresponding symbol. Inclusion and intersection between classes is defined. Same symbol serves for intersection and "and". Comments and explanations include arithmetic examples as the fact that common multiples of 2 and 3 are multiples of 6, and vice versa.

Mirror image of the usual symbol for inclusion serves for "then" between statements. Example:

For logic equivalence between two statements is used .=.

Step by step, syntactic rules for symbolic expresions are established.

Third part deals with "what" whose syimbol is the mirror image of epsilon. It corresponds to the expresion "such that"

One of the statements of this part could be paraphrased as:

If A is a class, then all of the x such that x belong to the class A is equal to class A.

In this example can be seen the reason for which Peano states that "belong" and "such that" are  inverse each other.

In theis third part the intersection definition could be paraphrased as

if a and b are classes, then, x such that (x belongs to a, and x belongs to b) = a intersection b

A later statement is: What (or such that) is distributive with respect to the "and" connective.

Part four of this chapter deals with negation of a statement or the complement of a set. The corresponding symbol is a small horizontal and bold line.

Peano uses lower case letters for sets or statements, isdistinctly. This kind of ambiguity could be interpreted as an implicit reckoning to the Boolean algebra of sets and statemens.

Double negation (or double complementation) is stated as -(-a) = a

Also is stated that if a is a class, the complement of a is a class. And is stated that given two classes a, b, if a is included in b, then the complement of b is included in the complement of a.

 Fifth part deals with union between sets. The same symbol is used for union and for the connective "or" between statements in the sense and/or.

Sixth part deals with "clase nullo" which corresponds to our empty set. Symbolized by an inverted V. The following is a textual statement

“clase nullo”, indica clase de objeto commune ad omni clase a. Responde ad 0 de Arithmetica.

The followig statements in today´s notation are included also in the 6th part.

Part 7th deals with "equal" symbolized with the greek iota

First statement above can be paraphrased as

 ι x = the y such that y = x.

In this final part Peano clearly establishes the conceptual difference between an individual and the class which contains a single individual.
Also, it is stated that symbols iota and epsilon side by side means =.

Arithmetic (Chapter II of Formulario)

Peano is known mainly by the set of axioms related with  natural numbers.

 N₀ symbolizes the set of natural numbers {0, 1, 2, …}.
 a+ symbolizes the number after number a. 
 0, N₀, + (next one) are undefined concepts

The statement .3 above is known as induction axiom. This axiom is a powerful tool for proving theorems about natural numbers or any other set which can be put in a one to one correspondence with them-

Using the "next" concept all of the natural numbers can be defined:

1 = 0+, 2 = 1+, 3 = 2+, etc.

Addition between natural numbers can be inductively defined:

a + 0 = a

a + (b+) = (a + b)+

Also, a + 1 = a+, a + 2 = (a+)+, …

Known properties of addition between natural numbers (closure, associative, commutative, etc.) are proved by means of induction.

Multiplication between natural numbers is defined inductively too. For any natural numbers a, b, c,

a × 0 = 0

a × (b + 1) = (a × b) + a

As for addition, properties for multiplication like a × 1 = a are proved by induction.

The power of a number, expressed in the form of a binary operation is defined inductively too. The symbol for this operation is attributed to De Morgan.

The corresponding properties are proved by induction. Peano employs the usual notation to

(a + b)² = a² + 2ab + b²,

 (a + b + c)³ = a³ + b³ + c³ + …

An addition operation between a number and a class is defined, with the correspondent properties. For instance, N₁  = N₀ + 1 is the class of all the natural numbers from 1 on.

This concept is used to define additive order between natural numbeers. For instance, 7 is greater than 5 means that 7 is in the class of 5 + N₁. Some properties of this operation are not trivial:

Multiplicative order is defined in analogous manner. For instance, N₀ × 5 is the set {0, 5, 10, 15,…}. This concept will be useful in the study of quotient between natural numbers.

Subtraction between natural numbers is defined in terms of addition. Paraphrasing,

b – a is the natural number x such that x + a = b, provided that b belongs to  a + N₀.

Properties like the following can be proved:

(b + a) – a = b,

a – (b + c) = a – b – c, provided that a belongs to b + c + N₀ .

Quotient between natural numbers is defined in terms of multiplication. Paraphrasing, b/a is the number x such that x × a = b, provided that a belongs to N₁ and b belongs to N₁ × a.

Properties like the following can be proved:

 (a + b)/c = (a/c) + (b/c), provided that a, b belong to N₁ × c.

The arithmetic chapter ends with the following concepts and the associated properties:

Number of a class

Maximum of a class

Minimum of a class

Quotient and reminder

Number of figures of a number

Order for a number

The factorial of a number is defined inductively as follows:

                                                            0! = 1

                                                            (a + 1)! = a!  × (a + 1)

Greatest common factor and least common multiple

Prime numbers

The maximum power of a prime number  p which divides the number a, mp(p, a).

 Euler's  Φ function       

The next section, called Vocabulario II, is similar to Vocabulario I after the chapter on Logic. It follows the initial part of this section. Note the allusion to other languages.

Algebra (chapter III in Formulario)

There are 27 parts in this chapter. It aims to develop with rigour and extensively, various general subjects. They are listed next, whitout details:

Functions, operations, correspondence

Post functions

About belonging elements-classes

Definition of a function

Union and intersection as operators

Positive and negative numbers

Absolute value

Rational numbers

Rational relative numbers

Integer part

Proper fraction

Limit superior

Real numbers

Logarithms

Sum operator

Product operator

Difference about functions

Bernoulli numbers

Medium number between superior and inferior limits

Number of elements of a class

Superior and inferior limits

Intervals

Probability

Complex numbers

Determinants

Linear and complex function of order n

Algebraic roots

The chapter ends also with its Vocabulario, from which a part is included here:

Geometry (chapter IV in Formulario)

It consists of four parts: (1) Points and vectors. (2) Line and plane. (3) Quaternions. (4)Outer product

Part 1. It is stated  that the idea of point cannot be logically defined and it comes from the physical world. Three axioms are paraphrased as follows:

p is a class

p does exist

If a belongs to p then there is another point , different from a, which does not coincides with a.

It is defined a relation between four points, incluiding a pertinent illustration.

In the so called position geometry, Peano states that if points a, b, c, d  are not in the same line, then line ab is parallel with line cd and line ac is parallel with line bd. Also,  points a, b coincide with points c, d by means of a translation movement, using a mechanical term.

The relation a – b = c – d, is an undefined term, where a, b, c, d are points. There are six axioms concerning this relation and a theory based in them is developed. A vector is defined as difference between two points. Other definitions concern to scalar product, module of a vector and product between a rational or irrational number and a vector. Reference is made to a publication: Principii di Geometria, whose undefined ideas are point and segment.

Part two. It deals with lines and planes. Line passing by a and is parallel to a vector u. Plane determined by a point and two vectors. Projection on a line. Translation. Symmetry. Motor:indicates movement of rigid body. Product of two symmetries. Homothecy.Sine and cosine.Coordinates.

Part three. Quaternion. The following could give some idea about.

Part four. Outer product of vectors is symbolized with α. It is defined by means of a determinant

At the end there is, as usual, the corresponding Vocabulario.

Limits (chapter V of Formulario)

This chapter has eight parts. Parts 1 and 2 deal with definition of limit, superior and inferior limits of sequences, uniqueness of limit, constant function, series, infinite products, Newton's binomial, convergence, limits with complex numbers, limits and vectors, continuity of functions. In this chapter some illustrations related with the so called Peano curve with which any plane region could be filled.

Part 3 of this chapter deals extensively with number e. Part 4 deals with number π. Part 5  is about inverses of functions log, sin, cos, tan. Part 6 develops the concept of angle. Part 7 is about the concept of rotation. Part 8 is about continuing fractions.

Differential Calculus (chapter VI of formulario)

Topics developed in this chapter are: derivative, differential, mean value theorem, L'Hospital theorem, Bernoulli-Taylor theorem, Lagrange's theorem, curvature, incremental ratios, complex functions, tangent lines and planes, normal plane, osculator, linear differential equations, harmonic movement, partial derivatives, applications to physics.

Integral Calculus (chapter VII of formulario)

Topics developed in this chapter are: circumscribed polygons, integral of a function, mean value theorem, integral with limits, methods of integration, improper integrals, Euler´s integral, integrability conditions, area, volume.

Applications to geometry and complement (chapter VIII of Formulario)

This chapter reproduces Dr. G. Pagliero's "Teoría de curvas". It develops topics as parabolas, ellipses, hyperbolas, exponentials, catenary curves, spiral, cycloid, epicycloid, cardioid, etc.A complement is included about complex numbers, differential equations, eliptic integrals and Fourier series.

The Formulario consists in mainly symbolic text with  comments and historic annotations like the following.

The item of Formulario which this article is referred to has 459 pages.