Tricky Puzzles

Any Song Puzzle

The Puzzle:

A poor woman and a rich woman are talking about music.

The poor woman says she has studied music and can name a song with any name in it.

The rich woman says "OK, if you can find a song with my son's name in it, I will give you a thousand dollars. His name is Demarcus-Jabari."

The poor woman gives her answer and is instantly $1,000 richer.

What was her answer?

Solution:

"Happy Birthday."

Starter Puzzles

12 Days Of Christmas Puzzle 

The Puzzle:

According to the traditional song, on the first day of Christmas (25th December), my true love sent to me:

. A partridge in a pear tree

On the second day of Christmas (26th December), my true love sent to me THREE presents:

. Two turtle doves
. A partridge in a pear tree

On the third day of Christmas (27th December and so on) my true love sent to me SIX presents:

. Three French hens
. Two turtle doves
. A partridge in a pear tree

This carries on until the the twelfth day of Christmas, when my true love sends me:

Twelve drummers drumming
Eleven pipers piping
Ten lords a-leaping
Nine ladies dancing
Eight maids a-milking
Seven swans a-swimming
Six geese a-laying
Five gold rings
Four calling birds
Three French hens
Two turtle doves
A partridge in a pear tree

After the twelve days of Christmas are over, how many presents has my true love sent me altogether?

Solution:

Day by Day:
1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 presents

Which is really interesting when you think there are 365 days in a typical year!


A kind reader, Martin J. Fowler, found this interesting symmetry in the total number of each present:

12 Partridges (12 days x 1)
22 Turtle Doves (11 days x 2)
30 French Hens (10 days x 3)
36 Calling Birds (9 days x 4)
40 Gold Rings (8 days x 5)
42 Geese Laying (7 days x 6)
42 Swans swimming (6 days x 7)
40 Maids a Milking (5 days x 8)
36 Ladies dancing (4 days x 9)
30 Lords a Leaping (3 days x 10)
22 Pipers Piping (2 days x 11)
12 Drummers drumming (1 day x 12)

Shapes Puzzles

A Perfect Match Puzzle 

The Puzzle:

In this diagram 11 matches make 3 squares:



Your challenge is to move 3 matches to show 2 squares.

Solution:

Logic Puzzles

Apples and Friends Puzzle

The Puzzle:

You have a basket containing ten apples. You have ten friends, who each desire an apple. You give each of your friends one apple.

Now all your friends have one apple each, yet there is an apple remaining in the basket.

How?

Solution:

You give an apple each to your first nine friends, and a basket with an apple to your tenth friend.

Each friend has an apple, and one of them has it in a basket.


(Alternative answer: one friend already had an apple and put it in the basket.)

Maths Puzzles

A Hole New Board Game Puzzle

How can I cut the board into only two pieces so that they will fill the hole exactly?

Solution:

Mathematics in Nature

Patterns in nature

 Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

 In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Hungarian biologist Aristid Lindenmayer and French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.

 Mathematics, physics and chemistry can explain patterns in nature at different levels. Patterns in living things are explained by the biological processes of natural selection. Studies of pattern formation make use of computer models to simulate a wide range of patterns.

History

 Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. Plato    (c 427 – c 347 BC) — looking only at his work on natural patterns — argued for the existence of universals. He considered these to consist of ideal forms (εἶδος eidos: "form") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect mathematical circle.Pythagoras explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.Empedocles to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.

 In 1202, Leonardo Fibonacci (c 1170 – c 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci.Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population.In 1917, D'Arcy Wentworth Thompson (1860–1948) published his book On Growth and Form. His description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants, is classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns and mollusc shells.

 The Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams.

 The German psychologist Adolf Zeising (1810–1876) claimed that the golden ratio was expressed in the arrangement of plant parts, in the skeletons of animals and the branching patterns of their veins and nerves, as well as in the geometry of crystals.

 Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their symmetry to support his faux-Darwinian theories of evolution.

 The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in 1885.

 In 1952, Alan Turing (1912–1954), better known for his work on computing and codebreaking, wrote The Chemical Basis of Morphogenesis, an analysis of the mechanisms that would be needed to create patterns in living organisms, in the process called morphogenesis.He predicted oscillating chemical reactions, in particular the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms can, Turing suggested, generate patterns of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis.

In 1968, the Hungarian theoretical biologist Aristid Lindenmayer (1925–1989) developed the L-system, a formal grammar which can be used to model plant growth patterns in the style of fractals.L-systems have an alphabet of symbols that can be combined using production rules to build larger strings of symbols, and a mechanism for translating the generated strings into geometric structures. In 1975, after centuries of slow development of the mathematics of patterns by Gottfried Leibniz, Georg Cantor, Helge von Koch, Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, crystallising mathematical thought into the concept of the fractal.

Causes

 Living things like orchids, hummingbirds, and the peacock's tail have abstract designs with a beauty of form, pattern and colour that artists struggle to match.^[18] The beauty that people perceive in nature has causes at different levels, notably in the mathematics that governs what patterns can physically form, and among living things in the effects of natural selection, that govern how patterns evolve.

 Mathematics seeks to discover and explain abstract patterns or regularities of all kinds.Visual patterns in nature find explanations in chaos theory, fractals, logarithmic spirals, topology and other mathematical patterns. For example, L-systems form convincing models of different patterns of tree growth.

 The laws of physics apply the abstractions of mathematics to the real world, often as if it were perfect. For example, a crystal is perfect when it has no structural defects such as dislocations and is fully symmetric. Exact mathematical perfection can only approximate real objects.Visible patterns in nature are governed by physical laws; for example, meanders can be explained using fluid dynamics.

 In biology, natural selection can cause the development of patterns in living things for several reasons, including camouflage, and different kinds of signalling, including mimicry and cleaning symbiosis.In plants, the shapes, colours, and patterns of insect-pollinated flowers like the lily have evolved to attract insects such as bees. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as nectar guides that can be seen at a distance.

Types of pattern

Symmetry

Symmetry is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids.Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies.

Among non-living things, snowflakes have striking sixfold symmetry: each flake's structure forming a record of the varying conditions during its crystallisation, with nearly the same pattern of growth on each of its six arms.Crystals in general have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike quasicrystals).Rotational symmetry is found at different scales among non-living things including the crown-shaped splash pattern formed when a drop falls into a pond,and both the spheroidal shape and rings of a planet like Saturn.

Symmetry has a variety of causes. Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction. But animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialised with a mouth and sense organs (cephalisation), and the body becomes bilaterally symmetric (though internal organs need not be).More puzzling is the reason for the fivefold (pentaradiate) symmetry of the echinoderms. Early echinoderms were bilaterally symmetrical, as their larvae still are. Sumrall and Wray argue that the loss of the old symmetry had both developmental and ecological causes.

 Reference from Wikipedia.

Daily uses of Mathematics

10 Ways We Use Math Everyday

Math is a part of our lives, whether we clean the house, make supper or mow the lawn. Wherever you go, whatever you do, you are using math daily without even realizing it. It just comes naturally. Chatting on the cell phone Chatting on the cell phone is the way of communicating for most people nowadays. It's easy, accessible and cost effective. Every one has a cell phone and it requires a basic knowledge of skill and math. You need to know numbers and how they work, and with today's technology you can do basically everything on your cell phone, from talking and faxing to surfing the Internet. In the kitchen Baking and cooking requires some mathematical skill as well. Every ingredient has to be measured and sometimes you need to multiply or divide to get the exact amount you need. Whatever you do in the kitchen requires math. Even just using the stove is basic math skills in action. Gardening Even doing something as mundane as gardening requires a basic math skill. If you need to plant or sow new seeds or seedlings you need to make a row or count them out or even make holes. So even without thinking you are doing math. Measuring skills is always needed, and calculations of the essence when doing something new in the garden. Arts When doing any form of art you are using math. Whether you're a sculptor, a painter, a dancer or even just doing a collage for fun, you will need to be able to measure, count and apply basic math to it. Every form of art is co-dependant upon math skills. Keeping a diary Keeping a diary has become an essential part of our daily lives. We run from place to place and appointment to appointment. Making appointments and having a time schedule that works for you requires math. Without a diary we will crash and burn. Some people even have to make appointments to take some time out. Math is a much needed skill in today's life. Planning an outing Every outing you plan needs your math skill. Whether you go to the beach or the zoo is irrelevant. You will plan your way there and you will use your time wisely, math is your guide that will assist you and help you. When driving you need fuel, oil and water, without it your car will break down. All of these require math. Banking Can you imagine going to the bank and not having any idea what you need to do or how to manage your finances. This will cause a huge disaster in your life, and you will be bankrupt within hours. Planning dinner parties How about that inevitable dinner party or cocktail that you have to host. Planning is essential, how many guests are attending, what foods are you serving, the ambience of the place where you want to host it and so many other essentials all requiring multiplication, division and subtraction. Decorating your home Whether you are painting, doing the flooring or just acquiring new furniture, you need math to make your sums add up. Everything you do inside or outside of your home needs math skills. From accessories to a new swimming pool and putting in new lighting. Statistics Every basic thing we use in life consist of history. That means statistics. Taking into account the past and the future, and keeping record of what has been done. Without statistics we won't know what worked and what didn't. It helps us to find balance and structure.

Maths Puzzles

If we put the numeral 1 at the beginning, we get a number three times smaller than if we put the numeral 1 at the end of the number.

Our Solution:

We can make an equation:

3(100000 + x) = 10x+1

(Why? Well, adding 100000 puts a 1 at the front of a five-digit number, and multiplying by 10 and adding 1 puts a 1 at the end of a number)

Solving this gives:

10x+1 = 3(100000 + x)
10x+1 = 300000 + 3x
10x = 299999 + 3x
7x = 299999
x = 299999/7 = 42857

The answer is 42857 (142857 is three times smaller than 428571)

Types of Numbers

                                   
                                                               Amicable Numbers

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.

The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.

The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992).(sequence A259180 in the OEIS). (Also see  A002025 and  A002046)

History

Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra (826–901). Other Arab mathematicians who studied amicable numbers are al-Majriti (died 1007), al-Baghdadi (980–1037), and al-Fārisī (1260–1320). The Iranian mathematician Muhammad Baqir Yazdi (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes.Much of the work of Eastern mathematicians in this area has been forgotten.

Thābit ibn Qurra's formula was rediscovered by Fermat (1601–1665) and Descartes (1596–1650), to whom it is sometimes ascribed, and extended by Euler (1707–1783). It was extended further by Borho in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs.The second smallest pair, (1184, 1210), was discovered in 1866 by a then teenage B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.

By 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a given bound, this bound being extended from 10⁸ in 1970, to 10¹⁰ in 1986, 10¹¹ in 1993, 10¹⁷ in 2015, and to 10¹⁸ in 2016.

As of April 2016, there are over 1,000,000,000 known amicable pairs.

Rules for generation

While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.

It states that if

p = 3×2^{n − 1} − 1,

q = 3×2ⁿ − 1,

r = 9×2^{2n − 1} − 1,

where n > 1 is an integer and p, q, and r are prime numbers, then 2ⁿ×p×q and 2ⁿ×r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n = 2, (17296, 18416) for n = 4, and (9363584, 9437056) for n = 7, but no other such pairs are known. Numbers of the form 3×2ⁿ − 1 are known as Thabit numbers. In order for Ibn Qurra's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of n.

To establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a natural integer. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.

Regular pairs

Let (m, n) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair (m, n) is said to be regular (see sequence A215491 in OEIS), otherwise it is called irregular or exotic. If (m, n) is regular and M and N have i and jprime factors respectively, then (m, n) is said to be of type (i, j).

For example, with (m, n) = (220, 284), the greatest common divisor is 4 and so M = 55 and N = 71. Therefore, (220, 284) is regular of type (2, 1).

Twin amicable pairs

An amicable pair (m, n) is twin if there are no integers between m and n belonging to any other amicable pair (sequence A273259 in the OEIS).

 Reference from wikipedia

Types of Number

                                                                Perfect Number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ₁(n) = 2n.

This definition is ancient, appearing as early as Euclid's Elements where it is called (perfect, ideal, or complete number). Euclid also proved a formation rule whereby ${\displaystyle q(q+1)/2}$ is an even perfect number whenever ${\displaystyle q}$ is a prime of the form ${\displaystyle 2^{p}-1}$ for prime ${\displaystyle p}$—what is now called a Mersenne prime. Much later, Euler proved that all even perfect numbers are of this form.This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

Examples

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in the OEIS).

History

In about 300 BC Euclid showed that if 2^p−1 is prime then (2^p−1)2^p−1 is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus had noted 8128 as early as 100 AD. Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000.St Augustine defines perfect numbers in City of God (Part XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336, 8,589,869,056 and 137,438,691,328) and listed a few more which are now known to be incorrect. In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest European reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time. In 1588, the Italian mathematician Pietro Cataldi also identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.

Even perfect numbers

Euclid proved that 2^p−1(2^p − 1) is an even perfect number whenever 2^p − 1 is prime 

For example, the first four perfect numbers are generated by the formula 2^p−1(2^p − 1), with p a prime number, as follows:

for p = 2:   2¹(2² − 1) = 6

for p = 3:   2²(2³ − 1) = 28

for p = 5:   2⁴(2⁵ − 1) = 496

for p = 7:   2⁶(2⁷ − 1) = 8128.

Prime numbers of the form 2^p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2^p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2^p − 1 with a prime p are prime; for example, 2¹¹ − 1 = 2047 = 23 × 89 is not a prime number.In fact, Mersenne primes are very rare—of the 9,592 prime numbers p less than 100,000,2^p − 1 is prime for only 28 of them.

Nicomachus (60-120 AD) conjectured that every perfect number is of the form 2^p−1(2^p − 1) where 2^p − 1 is prime. Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of that form. It was not until the 18th century that Leonhard Euler proved that the formula 2^p−1(2^p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem. As of January 2016, 49 Mersenne primes are known, and therefore 49 even perfect numbers (the largest of which is 2^74207280 × (2^74207281 − 1) with 44,677,235 digits).

An exhaustive search by the GIMPS distributed computing project has shown that the first 44 even perfect numbers are 2^p−1(2^p − 1) for

p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, and 32582657 (sequence A000043 in the OEIS).

Five higher perfect numbers have also been discovered, namely those for which p = 37156667, 42643801, 43112609, 57885161, and 74207281, though there may be others within this range. It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.

As well as having the form 2^p−1(2^p − 1), each even perfect number is the (2^p − 1)th triangular number (and hence equal to the sum of the integers from 1 to 2^p − 1) and the 2^p−1thhexagonal number. Furthermore, each even perfect number except for 6 is the ((2^p + 1)/3)th centered nonagonal number and is equal to the sum of the first 2^(p−1)/2 odd cubes:

${\displaystyle {\begin{aligned}6&=2^{1}(2^{2}-1)&&=1+2+3,\\[8pt]28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7=1^{3}+3^{3},\\[8pt]496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3},\\[8pt]8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3},\\[8pt]33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}.\end{aligned}}}$

Even perfect numbers (except 6) are of the form

${\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}}$

with each resulting triangular number (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with 3, 55, 903, 3727815, .... This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2^p−1(2^p − 1) with odd prime p and, in fact, with all numbers of the form 2^m−1(2^m − 1) for odd integer (not necessarily prime) m.

Owing to their form, 2^p−1(2^p − 1), every even perfect number is represented in binary as p ones followed by p − 1  zeros:

6₁₀ = 110₂

28₁₀ = 11100₂

496₁₀ = 111110000₂

8128₁₀ = 1111111000000₂

33550336₁₀ = 1111111111111000000000000₂.

Thus every even perfect number is a pernicious number.

Note that every even perfect number is also a practical number

Odd perfect numbers

It is unknown whether there is any odd perfect number, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists. More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.

Any odd perfect number N must satisfy the following conditions:

• N > 10¹⁵⁰⁰.
• N is not divisible by 105.
• N is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).
• N is of the form

${\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}$

where:

• q, p₁, ..., p_k are distinct primes (Euler).
• q ≡ α ≡ 1 (mod 4) (Euler).
• The smallest prime factor of N is less than (2k + 8) / 3.
• Either q^α > 10⁶², or p _j^2e_j  > 10⁶² for some j.
• N < 2^4k+1.

• The largest prime factor of N is greater than 10⁸.
• The second largest prime factor is greater than 10⁴, and the third largest prime factor is greater than 100.
• N has at least 101 prime factors and at least 10 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.

In 1888, Sylvester stated:

...a prolonged meditation on the subject has satisfied me that the existence of any one such [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.

Minor results

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

• The only even perfect number of the form x³ + 1 is 28 (Makowski 1962).
• 28 is also the only even perfect number that is a sum of two positive integral cubes (Gallardo 2010).
• The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, ${\displaystyle \sigma _{1}(n)=2n}$, and divide both sides by n):
  • For 6, we have ${\displaystyle 1/6+1/3+1/2+1/1=2}$;
  • For 28, we have ${\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}$, etc.
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
  • From these two results it follows that every perfect number is an Ore's harmonic number.
• The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form ${\displaystyle 2^{n-1}(2^{n}+1)}$ formed as the product of a Fermat prime ${\displaystyle 2^{n}+1}$ with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.
• The number of perfect numbers less than n is less than ${\displaystyle c{\sqrt {n}}}$, where c > 0 is a constant.In fact it is ${\displaystyle o({\sqrt {n}})}$, using little-o notation.
• Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1, base 9.Therefore in particular the digital root of every even perfect number other than 6 is 1.
• The only square-free perfect number is 6.

     

   Reference from wikipedia