Which is the smallest perfect number

2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. There is a 2 p − 1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of p = 2, 2 2 − 1 = 3 is prime, and 2 2 − 1 × (2 2 − 1) = 2 × 3 = 6 is perfect. It is currently an x is ( e γ / log 2) × log log x, where e is γ is log is the 10 1500. The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2023 The displayed ranks are among indices currently known as of 2022 p = 57,885,161 have been checked and verified as of October2021 name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last 6 digits of each number are shown. Table of all 51 currently-known Mersenne primes and corresponding perfect numbers Rank p Mersenne prime Mersenne prime digits Perfect number Perfect number digits Discovery Discoverer Method Ref. 1 2 1 1 Ancient times Known to Unrecorded 2 3 1 2 3 5 2 3 4 7 3 4 5 13 8191 4 33550336 8 c. 1456 Anonymous 6 17 131071 6 858986...

• • • • • • What Is a Perfect Number? Perfect numbers in math are the numbers (positive integers) that can be expressed as the sum of their factors (excluding the number itself). Factors of a number are the numbers that divide a given number exactly without leaving a remainder. A factor of a number is always less than or equal to the number. Perfect Numbers Definition A perfect number is defined as a positive integer that can be expressed as the sum of its proper factors (factors except for the number itself). Perfect number examples : 6, 28, 496 The factors of 6 are 1, 2, 3 and 6. We can write $6 = 1 + 2 + 3$. The smallest perfect number is 6. When the sum of all the divisors of a number is equal to twice the number, the number is known as complete number. All the perfect numbers are also complete numbers. Theorem of Even Perfect Numbers An even natural number N is perfect if and only if it is of the form $N = 2^\;(3) = 6$ …a perfect number How to Find a Perfect Number According to Euclid, there is an expression that can be used to check if a number is a perfect number. Around 2000 years ago, Euclid showed that any even perfect number can be represented by, If $(2^n\;-\;1)$ is a prime number, then $2^ \times (2^n\;-\; 1)$ 1 1 1 – 2 2 3 (prime number) 6 (perfect number) 3 4 7 (prime number) 28 (perfect number) 4 8 15 (not a prime number) – 5 16 31 (prime number) 496 (perfect number) 6 32 63 (not a prime number) – 7 64 127 (prime number) 8128 (perfect number) 8 128 255 (not...

Perfect Numbers Perfect Numbers A perfect number is a positive integer that is equal to the sum of all its proper divisors. (A proper divisor of n is a number that exactly divides n, but it is not n itself.) For example, the smallest perfect number is 6, whose proper divisors are 1, 2 , and 3; indeed 6 = 1 + 2 + 3. The next perfect number is 28 (its proper divisors are 1, 2, 4, 7, 14, which sum to 28). The next two perfect numbers are 496 and 8128. Beyond that the values get really huge. Only about 10 perfect numbers are known before 1900. And even now only 38 of them have been discovered. It is not yet known if there are infinitely many perfect numbers. All of the 38 perfect numbers discovered so far are even numbers. That is, since the subject was first studied by ancient Greeks over 2000 years ago, not a single odd number has been found to be a perfect number. Indeed, all odd numbers up to 10 300 (at least) have been checked without finding any perfect number among them. Whether or not odd perfect numbers exist is another problem unsolved to this day. As early as during Euclid's time, it was known that if a number of the form 2 n - 1 is a prime, then the product 2 n-1(2 n - 1) is an even perfect number. Later, Euler proved that all even perfect numbers are of such form. Recall that a prime of the form 2 n - 1 is called a Mersenne Prime (the first few are 3, 7, 31, and 127). Therefore, the fact that 2 n - 1 is a prime is a necessary and sufficient condition that 2 n-1(2 ...

Euclid, from Rafael’s “School of Athens”, Vatican Museum, Rome, photo courtesy Clay Mathematics Institute Perfect numbers A perfect number is a positive integer whose divisors (not including itself) add up to the integer. The smallest perfect number is $6$, since $6 = 1 + 2 + 3$. The next is $28 = 1 + 2 + 4 + 7 + 14$, followed by $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$ and $8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64$ $+ 127 + 254 + 508 + 1016 + 2032 + 4064$. The notion of a perfect number is at least 2300 years old. In approximately 300 BCE, Euclid showed (using modern notation) that if $2^p – 1$ is a prime number (which implies, by the way, that $p$ itself is prime), then $2^(−22017975903) = \sigma(3^4) \sigma(7^2) \sigma(11^2) \sigma(19^2) \sigma(-127)$$ $$= (1 + 3 + 3^2 + 3^3 + 3^4) (1 + 7 + 7^2) (1 + 11 + 11^2) (1 + 19 + 19^2) (1 + (-127)) = -44035915806 = 2 \cdot (−22017975903).$$ But again, this calculation is invalid, in this case because the integer in question is negative, and $\sigma(-127) = -128$ not $-126$ as used above. But in both cases, these “spoofs” are definitely interesting, and possibly might inspire a line of attack to certify that OPNs simply cannot exist. Computer searches The most recent Why the interest in spoofs? According to Nielsen, Any behavior of the larger set has to hold for the smaller subset. … So if we find any behaviors of spoofs that do not apply to the more restricted class, we can automatically rule out the possibility of an OPN. So...

A Mersenne prime is a prime number that can be written in the form \(2^-1=2047\) isn't a prime number. Lucas–Lehmer primality test is a primality test (an algorithm for determining whether an input number is prime) for Mersenne primes. It's currently known to be the most efficient test for Mersenne primes. First we start with \(n=0\). Then \(-1\) is not prime. \(_\square\) There are \(51\) known Mersenne primes, with the largest known prime being \[ \large 2^ - 1,\] which is over 24 million digits long! This enormous number was discovered by Patrick Laroche in 2018, as part of the GIMPS (Great Internet Mersenne Prime Search). It is a collaborative effort to find new primes by pooling computing power online. It has 24,862,048 digits in total.

A perfect number is a positive integer that equals the sum of its proper divisors, that is, positive divisors excluding the number itself. For example, \(6\) is a perfect number because the proper divisors of \(6\) are \(1,2,\) and \(3,\) and \(6=1+2+3.\) The sum of all positive divisors of a number \(n\) is denoted by \(\sigma(n)\). A perfect number is therefore a positive integer \( n \) such that \( \sigma(n) = 2n.\) Perfect numbers were of great interest to ancient mathematicians including the Greeks, who attributed mystical significance to the property. It is perhaps somewhat surprising that many elementary questions about perfect numbers are still open. Most notably, it is not known whether there are infinitely many perfect numbers, and it is not known whether there are any odd perfect numbers. Euclid proved in the Elements that if \( 2^p-1 \) is prime, then \( 2^\] But note that the right side is at least \( c + (2^a-1)c \) since these are two divisors of \( (2^a-1)c \), and those two divisors already add up to \( 2^ac \). So the conclusion is that equality holds and these are the only two divisors of \( (2^a-1)c \). This can only happen if \( c = 1 \) and \( 2^a-1 \) is prime. This completes the proof. \(_\square \) \((\)Exercise: Where did the proof use the fact that \( a \ge 2?) \) The following is a list of some interesting properties of perfect numbers. Unlike many lists in the literature, this list is confined to nontrivial properties: for instance, the fact t...

Perfect Numbers What are the Perfect Numbers? Definition: A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128. A Perfect Number “n”, is a positive integer which is equal to the sum of its factors, excluding “n” itself. Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by, N = 2 p-1(2 p -1) where p is a prime for which 2 p -1 is a Mersenne prime. That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2 p -1 is Perfect Number Table: The following gives a table of the first nine Mersenne Primes and Perfect Numbers Prime, p Mersenne Prime, 2 p -1 Perfect Number, 2 p-1(2 p -1) 2 3 6 3 7 28 5 31 496 7 127 8128 13 8191 33550336 17 131071 8589869056 19 524287 137438691328 31 2147483647 2305843008139952128 61 2305843009213693951 2658455991569831744654692615953842176 History of Perfect Number It is not known when Perfect Numbers were first discovered, or when they were studied, it is thought that they may even have been known to the Egyptians, and may have even been known before. Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers (O’Connor and Robertson, 2004). The Pythagoreans found the number 6 interesting (more for its mystical and num...

2-perfection of the number 6 In multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a For a given k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive n (the σ( n)) is equal to kn; a number is thus 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11. It is unknown whether there are any 1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... (sequence Example [ ] The sum of the divisors of 120 is 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360 which is 3 × 120. Therefore 120 is a 3-perfect number. Smallest known k-perfect numbers [ ] The following table gives an overview of the smallest known k-perfect numbers for k ≤ 11 (sequence k Smallest known k-perfect number Factors Found by 1 ancient 2 2 × 3 ancient 3 2 3 × 3 × 5 ancient 4 30240 2 5 × 3 3 × 5 × 7 5 14182439040 2 7 × 3 4 × 5 × 7 × 11 2 × 17 × 19 René Descartes, circa 1638 6 154345556085770649600 (21 digits) 2 15 × 3 5 × 5 2 × 7 2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 7 141310897947438348259849402738485523264343544818565120000 (57 digits) 2 32 × 3 11 × 5 4 × 7 5 × 11 2 × 13 2 × 17 × 19 3 × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479 TE Mason, 1911 8 826809968707776137289924...057256213348352000000000 (133 digits) 2 62 × 3 15 × 5 9 ×...

20 number of rooted trees with 6 vertices number of faces of an icosahedron and vertices of a dodecahedron number of quarter or half turns required to optimally solve a Rubik’s cube in the worst case base of the ancient Mayan number system sum of the first 4 triangular numbers, and thus a tetrahedral number