.]]>

In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).

**Examples**

The first prime triplets are

(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887)

**Subpairs of primes**

A prime triplet contains a single pair of:

* Twin primes: (p, p + 2) or (p + 4, p + 6);

* Cousin primes: (p, p + 4) or (p + 2, p + 6); and

* Sexy primes: (p, p + 6).

**Higher-order versions**

A prime can be a member of up to three prime triplets - for example, 103 is a member of (97, 101, 103), (101, 103, 107) and (103, 107, 109). When this happens, the five involved primes form a prime quintuplet.

A prime quadruplet (p, p + 2, p + 6, p + 8) contains two overlapping prime triplets, (p, p + 2, p + 6) and (p + 2, p + 6, p + 8).

]]>**Gist**

The brute force approach is a guaranteed way to find the correct solution by listing all the possible candidate solutions for the problem. It is a generic method and not limited to any specific domain of problems. The brute force method is ideal for solving small and simpler problems.

**Summary**:

**Algorithm**

A brute force algorithm is a simple, comprehensive search strategy that systematically explores every option until a problem’s answer is discovered. It’s a generic approach to problem-solving that’s employed when the issue is small enough to make an in-depth investigation possible. However, because of their high temporal complexity, brute force techniques are inefficient for large-scale issues.

**Key takeaways:**

Methodical Listing: Brute force algorithms investigate every potential solution to an issue, usually in an organized and detailed way. This involves attempting each option in a specified order.

Relevance: When the issue space is small and easily explorable in a fair length of time, brute force is the most appropriate method. The temporal complexity of the algorithm becomes unfeasible for larger issue situations.

Not using optimization or heuristics: Brute force algorithms don’t use optimization or heuristic approaches. They depend on testing every potential outcome without ruling out any using clever pruning or heuristics.

**Features of the brute force algorithm**

* It is an intuitive, direct, and straightforward technique of problem-solving in which all the possible ways or all the possible solutions to a given problem are enumerated.

* Many problems are solved in day-to-day life using the brute force strategy, for example, exploring all the paths to a nearby market to find the minimum shortest path.

* Arranging the books in a rack using all the possibilities to optimize the rack spaces, etc.

* Daily life activities use a brute force nature, even though optimal algorithms are also possible.

PROS AND CONS OF BRUTE FORCE ALGORITHM:

Pros:

* The brute force approach is a guaranteed way to find the correct solution by listing all the possible candidate solutions for the problem.

* It is a generic method and not limited to any specific domain of problems.

* The brute force method is ideal for solving small and simpler problems.

* It is known for its simplicity and can serve as a comparison benchmark.

Cons:

* The brute force approach is inefficient. For real-time problems, algorithm analysis often goes above the O(N!) order of growth.

* This method relies more on compromising the power of a computer system for solving a problem than on a good algorithm design.

* Brute force algorithms are slow.

* Brute force algorithms are not constructive or creative compared to algorithms that are constructed using some other design paradigms.

**Conclusion**

Brute force algorithm is a technique that guarantees solutions for problems of any domain helps in solving the simpler problems and also provides a solution that can serve as a benchmark for evaluating other design techniques, but takes a lot of run time and inefficient.

**Details**

**Proof of exhaustion**

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages:

* A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases.

* A proof of each of the cases.

The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.

In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching.

**Example**

Proof by exhaustion can be used to prove that if an integer is a perfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9.

**Proof:**

Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three cases are exhaustive:

Case 1: If n = 3p, then

, which is a multiple of 9.Case 2: If , then , which is 1 more than a multiple of 9. For instance, if n = 4 then .

Case 3: If , then , which is 1 less than a multiple of 9. For instance, if n = 5 then . Q.E.D.

**Elegance**

Mathematicians prefer to avoid proofs by exhaustion with large numbers of cases, which are viewed as inelegant. An illustration as to how such proofs might be inelegant is to look at the following proofs that all modern Summer Olympic Games are held in years which are divisible by 4:

Proof: The first modern Summer Olympics were held in 1896, and then every 4 years thereafter (neglecting exceptions such as when the games were not held due to World War I and World War II along with the 2020 Tokyo Olympics being postponed to 2021 due to the COVID-19 pandemic.). Since 1896 = 474 × 4 is divisible by 4, the next Olympics would be in year

, which is also divisible by four, and so on (this is a proof by mathematical induction). Therefore, the statement is proved.The statement can also be proved by exhaustion by listing out every year in which the Summer Olympics were held, and checking that every one of them can be divided by four. With 28 total Summer Olympics as of 2016, this is a proof by exhaustion with 28 cases.

In addition to being less elegant, the proof by exhaustion will also require an extra case each time a new Summer Olympics is held. This is to be contrasted with the proof by mathematical induction, which proves the statement indefinitely into the future.

**Number of cases**

There is no upper limit to the number of cases allowed in a proof by exhaustion. Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving a chess endgame puzzle might involve considering a very large number of possible positions in the game tree of that problem.

The first proof of the four colour theorem was a proof by exhaustion with 1834 cases.[4] This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

In general the probability of an error in the whole proof increases with the number of cases. A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection. Other types of proofs—such as proof by induction (mathematical induction)—are considered more elegant. However, there are some important theorems for which no other method of proof has been found, such as

* The proof that there is no finite projective plane of order 10.

* The classification of finite simple groups.

* The Kepler conjecture.

* The Boolean Pythagorean triples problem.

**Gist**

Closure property states that any operation conducted on elements within a set gives a result which is within the same set of elements. Integers are either positive, negative or zero. They are whole and not fractional. Integers are closed under addition.

**Summary**

The closure property means that a set is closed for some mathematical operation. That is, a set is closed with respect to that operation if the operation can always be completed with elements in the set. Thus, a set either has or lacks closure with respect to a given operation.

For example, the set of even natural numbers, [2, 4, 6, 8, . . .], is closed with respect to addition because the sum of any two of them is another even natural number, which is also a member of the set. (Natural numbers are defined as the set: [1, 2, 3, 4, . . .].) It is not closed with respect to division because the quotients 6/2 and 4/8, for example, cannot be computed without using odd numbers (6/2 = 3) or fractions (4/8 = ½;), which are not members of the set.

Knowing the operations for which a given set is closed helps one understand the nature of the set. Thus, one knows that the set of natural numbers is less versatile than the set of integers because the latter is closed with respect to subtraction, but the former is not. (Integers are defined as the set: [. . .-3, -2, -1, 0, 1, 2, 3, . . .].) Similarly one knows that the set of polynomials is much like the set of integers because both sets are closed under addition, multiplication, negation, and subtraction, but are not closed under division.

Particularly interesting examples of closure are the positive and negative numbers. In mathematical structure, these two sets are indistinguishable except for one property, closure with respect to multiplication. Once one decides that the product of two positive numbers is positive, the other rules for multiplying and dividing various combinations of positive and negative numbers follow. Then, for example, the product of two negative numbers must be positive, and so on.

The lack of closure is one reason for enlarging a set. For example, without augmenting the set of rational numbers with the irrationals, one cannot solve an equation such as x2 = 2, which can arise from the use of the pythagorean theorem. Without extending the set of real numbers to include imaginary numbers, one cannot solve an equation such as x^2 + 1 = 0, contrary to the fundamental theorem of algebra.

Closure can be associated with operations on single numbers as well as operations between two numbers. When the Pythagoreans discovered that the square root of 2 was not rational, they had discovered that the rationals were not closed with respect to taking roots.

Although closure is usually thought of as a property of sets of ordinary numbers, the concept can be applied to other kinds of mathematical elements. It can be applied to sets of rigid motions in the plane, to vectors, to matrices, and to other things. For instance, one can say that the set of three-by-three matrices is closed with respect to addition.

Closure, or the lack of it, can be of practical concern, too. Inexpensive, four-function calculators rarely allow the user to use negative numbers as inputs. Nevertheless, if one subtracts a larger number from a smaller number, the calculator will complete the operation and display the negative number that results. On the other hand, if one divides 1 by 3, the calculator will display 0.333333, which is close, but not exact. If an operation takes a calculator beyond the numbers it can use, the answer it displays will be.

**Details**

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 - 2 is not a natural number, although both 1 and 2 are.

Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.

The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set.

**Definitions**

Let S be a set equipped with one or several methods for producing elements of S from other elements of S. A subset X of S is said to be closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the closure property.

The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set. It follows that for every subset Y of S, there is a smallest closed subset X of S such that

(it is the intersection of all closed subsets that contain Y). Depending on the context, X is called the closure of Y or the set generated or spanned by Y.The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in

a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the Zariski closure of a set V of points is the smallest algebraic set that contains V.**In algebraic structures**

An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers

In this context, given an algebraic structure S, a substructure of S is a subset that is closed under all operations of S, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as S. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.

Given a subset X of an algebraic structure S, the closure of X is the smallest substructure of S that is closed under all operations of S. In the context of algebraic structures, this closure is generally called the substructure generated or spanned by X, and one says that X is a generating set of the substructure.

For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element. Here, the auxiliary operations are the nullary operation that results in the identity element and the unary operation of inversion. A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non-empty. So, a non-empty subset of a group that is closed under multiplication and inversion is a group that is called a subgroup. The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group.

In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.

Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology. For example, in a commutative ring, the closure of a single element under ideal operations is called a principal ideal.

]]>The best -

The three Is - Intuition, Intelligence, and Improvement - I think that is sufficient in Mathematics.

]]>I'm seeing a lot of LaTex errors in this. I hope you don't mind but I've had a go at editing some but I'm worried I might have changed your proof in the process. Please have a look and see if it's ok. Also ask if you are unsure of the Latex that works for MIF. Not all feaures of the coding do, I'm sorry. What did you want list to do? Some frac errors towards the end.

Bob

]]>In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though

and the left-exponent are common.Under the definition as repeated exponentiation,

means , where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation n-1 times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

Tetration is also defined recursively as

allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.

The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.

Tetration is used for the notation of very large numbers.

**Introduction**

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as

, is considered to be the zeroth operation.**Addition**

n copies of 1 added to a.

**Multiplication**

n copies of a combined by addition.

**Exponentiation**

n copies of a combined by multiplication.

**Tetration**

n copies of a combined by exponentiation, right-to-left.

Succession, (a′ = a + 1), is the most basic operation; while addition (a + n) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of n successors of a; multiplication (a × n) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving n numbers of a. Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration

as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.The parameter a is referred to as the base, while the parameter n may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real

and non-negative integer , we can define recursively as:The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as

, , and as well – many of these extensions are areas of active research.]]>b) 12 is the first sublime number.

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers: 12 and

. The second of these has 76 decimal digits:6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.

c) We know 6 is the first perfect number : Sum of the factors whose proper factors sum to the number itself.

(1 + 2 + 3 = 6).

28 is the second perfect number.

496 is the third perfect number.

8128 is the fourth perfect number.

d) 17 is the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.

e) 25 is the first centered square number besides 1 that is also a square number.

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

f) 30 is the smallest sphenic number.

In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers.

Definition : A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.

Examples : The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...

g) 142857 is the smallest base 10 cyclic number.

A cyclic number is an integer in which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

142857 × 1 = 142857

142857 × 2 = 285714

142857 × 3 = 428571

142857 × 4 = 571428

142857 × 5 = 714285

142857 × 6 = 857142.

h) 9814072356 is the largest perfect power that contains no repeated digits in base ten.

i) Pandigital number: In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689.

]]>To be sure I understand I'm going to apply what you said first to the related case of

Taking logs of both sides we get:

Rearranging:

If we now make the y=x^t substitution:

And since y=x^t:

We now have a way to generate solutions. When I try to use this method to find those more complicated answers I began listing earlier, however, I still don't know how, other than by graphing it (which is very difficult for negative answers). For example, graphically it seems there must be some t the real part of which is around -2.56 for which

, yielding the solution to the original problem y=2, x=-0.767... . Is this what happens when you consider t over all the complex numbers? I can see now the solution space for positive t is rather simple (for both equations), but it's still tricky to assess when t is negative. I suppose I am still uninformed, and will come back in a little bit. ]]>In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.

Various notations have been used to represent hyperoperations. One such notation is

. Another notation is , an infix notation which is convenient for ASCII. The notation is known as 'square bracket notation'.Knuth's up-arrow notation

is an alternative notation. It is obtained by replacing in the square bracket notation by arrows.For example:

the single arrow

represents exponentiation (iterated multiplication)the double arrow represents tetration (iterated exponentiation)

the triple arrow represents pentation (iterated tetration)

The general definition of the up-arrow notation is as follows (for :

Here, stands for n arrows, so for example.]]>

So I want something from you first:

Using a graph plotter, try a = 4 then 5 then 1 then 1/2 then 1/3.

Comment on what you are observing in these cases. Make three conclusions about how the value of a affects the graph.

Then use the series expansion for e^x with x = 1 to compute e to 8 decimal places .

Post the steps in your working and your final result. Pay particular attention to explaining how you know your answer is accurate to 8 dp (without just comparing with a published result).

Bob

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