The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term.
What is a conjecture in math?
Conjectures A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases.
What is the Poincaré conjecture in geometry?
Poincaré Conjecture: (proposed 1904 by Henri Poincaré, proved 2002 by Grigori Perelman) Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The Poincaré conjecture has been so recently proved that it is still popularly known as a conjecture rather than as the "Poincaré theorem."
Who coined the term “conjecture”?
Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is related to hypothesis, which in science refers to a testable conjecture. ^ "Definition of CONJECTURE". www.merriam-webster.com. Retrieved 2019-11-12.
What is a false conjecture?
Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. the Pólya conjecture and Euler's sum of powers conjecture).
How do you calculate conjecture?
2:518:02Making a Conjecture - YouTubeYouTubeStart of suggested clipEnd of suggested clipWell you can get a hint by what they're giving you the sum of two. Positive. Numbers okay so theMoreWell you can get a hint by what they're giving you the sum of two. Positive. Numbers okay so the thing is 3 53 35. And 6 they're all.
What is conjecture in math example?
A conjecture is a good guess or an idea about a pattern. For example, make a conjecture about the next number in the pattern 2,6,11,15... The terms increase by 4, then 5, and then 6. Conjecture: the next term will increase by 7, so it will be 17+7=24.
What is conjecture in sequence?
The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term.
What is a conjecture in algebra?
A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases.
How do you make and test a conjecture?
How to Have Students Make and Test Conjectures?Grab a student's attention by presenting them with a thought provoking research question.Engage the students by having them make a prediction(s) about possible outcomes to this question and then have them explain and share their reasoning.More items...•
What is conjecture reasoning?
A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.
How do you make a conjecture on a graph?
0:014:14Using a Trend Line to Make a Conjecture - YouTubeYouTubeStart of suggested clipEnd of suggested clipAll right how we're going to do that is by finding the linear equation of the line in slope-MoreAll right how we're going to do that is by finding the linear equation of the line in slope-intercept form y equals MX plus B. If we recall M is equal to slope and B is equal to the y-intercept.
Who invented math?
Archimedes is considered the Father of Mathematics for his significant contribution to the development of mathematics. His contributions are being used in great vigour, even in modern times.
What is a Conjecture?
Parents make conjectures all the time; without even realizing that they do, they form conclusions about their children. Susie notices that when she buys strawberry ice cream, her 3-year-old son Johnny always ask for seconds, but when she buys vanilla, he leaves some in the bowl. What conclusions do you think Susie would make? Of course, she would think that Johnny likes strawberry more than vanilla.
What is a conjecture in science?
A conjecture is like a hypothesis to a scientist. Scientists write hypotheses and test them to see if they are true. A conjecture is just an initial conclusion that you formed based on what you see and already know. Making a conjecture doesn't mean that you are correct or incorrect. All mathematical theorems began with a conjecture. Mathematicians notice a pattern in numbers or shapes, then they perform a number of operations and solve numerous equations to prove their conjecture.
How to disprove a conjecture?
All you need to disprove a conjecture is one example. That example is called a counterexample. Remember that 'counter' means to go against, so a counterexample is a statement used to disprove a conjecture.
Is it correct to make a conjecture?
Making a conjecture doesn't mean that you are correct or incorrect. All mathematical theorems began with a conjecture. Mathematicians notice a pattern in numbers or shapes, then they perform a number of operations and solve numerous equations to prove their conjecture.
Does making conjectures mean the conclusion is true?
Making conjectures doesn't mean that the conclusion is true.
Is 4 a prime number?
2 + 2 = 4, but 4 is not a prime number.
Do parents make conjectures about their children?
As we mentioned, parents also make conject ures about their child's health and well-being. If they notice something, they will make a few more observations and form some conclusions. Please note that forming a conjecture is only the first step, doing something about the conjecture to prove or disprove it is another step and has other names.
Who invented the term "conjecture"?
Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is related to hypothesis, which in science refers to a testable conjecture.
When is a conjecture considered proven?
A conjecture is considered proven only when it has been shown that it is logically impossible for it to be false. There are various methods of doing so; see methods of mathematical proof for more details.
What is the main conjecture of geometric topology?
The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.
What is the Weil conjecture?
Main article: Weil conjectures. In mathematics, the Weil conjectures were some highly influential proposals by André Weil ( 1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields .
What is Fermat's last theorem?
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers#N#a {displaystyle a}#N#,#N#b { displaystyle b}#N#, and#N#c { displaystyle c}#N#can satisfy the equation#N#a n + b n = c n { displaystyle a^ {n}+b^ {n}=c^ {n}}#N#for any integer value of#N#n { displaystyle n}#N#greater than two .
When was the four color theorem confirmed?
For example, the validity of the 1976 and 1997 brute-force proofs of the four color theorem by computer was initially doubted, but was eventually confirmed in 2005 by theorem-proving software. When a conjecture has been proven, it is no longer a conjecture but a theorem.
When was the color conjecture first proposed?
Möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of countries of England, noticed that only four different colors were needed.
How to conjecture closed forms?
Usually, when confronted with a sequence, the simplest way to conjecture some closed forms is to use the method of differences.
What happens if you apply the method of differences to your ( 1)?
Coming to your problems, if you apply this method to your ( 1) you notice something strange: the method of differences never terminates! After a while you get stuck in a loop and the differences are always the same. This tells us an important bit of information: the sequence cannot be expressed as a polynomial.
What does "conjecture" mean?
Definition of conjecture (Entry 2 of 2) transitive verb. 1 : to arrive at or deduce by surmise or guesswork : guess scientists conjecturing that a disease is caused by a defective gene. 2 : to make conjectures as to conjecture the meaning of a statement. intransitive verb.
Where does the word "conjecture" come from?
"Conjecture" derived via Middle English and Middle French from the Latin verb conicere ("to throw together"), a combination of "com-" ("together") and "jacere" ("to throw").
Is your plan based on conjecture?
Your plan is based on (nothing more than) conjecture. Most of the book is conjecture, not fact. The criminal's motive remains a matter of conjecture. [=people can only guess about the criminal's motive; no one knows the criminal's motive] Hide
Why do mathematicians think the conjecture is true?
Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.
What is the Collatz conjecture?
The Collatz conjecture states that all paths eventually lead to 1. The Collatz conjecture is a conjecture in mathematics that concerns sequences defined as follows : start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term.
How many cycles are there in Collatz conjecture?
Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of f. These cycles are listed here, starting with the well-known cycle for positive n :
What is the lower bound of 114 208 327 604?
This lower bound is consistent with the above result, since 114 208 327 604 = 17 087 915 × 361 + 85 137 581 × 1269.
How to find a positive integer?
Consider the following operation on an arbitrary positive integer : 1 If the number is even, divide it by two. 2 If the number is odd, triple it and add one.
Who used the precomputation to find a counterexample to the Collatz conjecture?
For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of n. If, for some given b and k, the inequality
Who proved that a natural generalization of the Collatz problem is algorithmically undecidable?
In 1972, John Horton Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable.
Overview
Important examples
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and can satisfy the equation for any integer value of greater than two.
This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica, where he claimed that he had a proof that was too large t…
Resolution of conjectures
Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, …
Conditional proofs
Sometimes, a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that — amongst other things — makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true. In fact, in anticipation of its eventual proof, some have even proceeded to develop further proofs which are contingent …
In other sciences
Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is related to hypothesis, which in science refers to a testable conjecture.
See also
• Bold hypothesis
• Futures studies
• Hypotheticals
• List of conjectures
• Ramanujan machine
External links
• Media related to Conjectures at Wikimedia Commons
• Open Problem Garden
• Unsolved Problems web site