Random word

Explore Collatz through 8 example sentences from English. Ideal for language learners, writers and word enthusiasts.

Rare word

Collatz in a sentence

Using Collatz

  • In the example corpus, collatz often appears in combinations such as: the collatz, collatz graph, collatz conjecture.

Context around Collatz

  • Average sentence length in these examples: 25.5 words
  • Position in the sentence: 2 start, 5 middle, 1 end
  • Sentence types: 8 statements, 0 questions, 0 exclamations

Often before:

standard 1×

Often after:

graph 3× conjecture 2× process 1× map 1× iteration 1×

Sentence patterns with Collatz

These short patterns come directly from the example sentences and show how the word is often embedded:

by the collatz process then 1× of the collatz graph generated 1× so called collatz graph 1× the collatz conjecture is 1× the collatz graph is 1× the standard collatz map can 1× to the collatz conjecture is 1× to the collatz iteration it 1×

Sentence openings and endings

Common sentence openings:

a probabilistic heuristic 1× extensions to larger 1× iterating with odd 1× other formulations of 1× rigorous bounds although 1× the collatz conjecture 1×

Common sentence endings:

all positive integers 1× bottom up fashion 1× called collatz graph 1× in lowest terms 1× just positive integers 1× numbers do so 1×

Words around Collatz

Just before it in the word index:

collations 4 collation 122 collatinus 6 collating 29 collates 15 collaterals 11

Just after it in the word index:

collazo 16 collboni 4 colle 10 colleage 2 colleague 1.445 colleagues 3.634

Similar in corpus frequency:

aargau 11 abacos 11 abboud 11 abbreviate 11 abdelkader 11 abdomens 11

Corpus analysis for Collatz

  • In this selection, "collatz" usually appears in the middle of the sentence. The average example has 25.5 words, and this corpus slice is mostly made up of statements.
  • Around the word, standard, graph, conjecture and process stand out and add context to how "collatz" is used.
  • Recognizable usage signals include by the collatz process then and of the collatz graph generated. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "collatz" sits close to words such as aargau, abacos and abboud, which helps place it inside the broader word index.

Example types with collatz

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

The Collatz conjecture is equivalent to the statement that all paths eventually lead to 1. Collatz1000mathematica. (16 words)

Other formulations of the conjecture In reverse The first 21 levels of the Collatz graph generated in bottom-up fashion. (20 words)

There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph. (21 words)

Longer example sentences

The Collatz graph is a graph defined by the inverse relation So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads to all positive integers. (35 words)

A probabilistic heuristic If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one. (31 words)

Iterating with odd denominators or 2-adic integers The standard Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. (30 words)

Example sentences (8)

A probabilistic heuristic If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one.

Extensions to larger domains Iterating on all integers An extension to the Collatz conjecture is to include all integers, not just positive integers.

Iterating with odd denominators or 2-adic integers The standard Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms.

Other formulations of the conjecture In reverse The first 21 levels of the Collatz graph generated in bottom-up fashion.

Rigorous bounds Although it is not known rigorously whether all positive numbers eventually reach one according to the Collatz iteration, it is known that many numbers do so.

The Collatz conjecture is equivalent to the statement that all paths eventually lead to 1. Collatz1000mathematica.

The Collatz graph is a graph defined by the inverse relation So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads to all positive integers.

There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph.

Common combinations with collatz

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "collatz" in a sentence?
An example: "A probabilistic heuristic If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one." This page contains 8 example sentences with the word "collatz" from authentic English texts.
How many example sentences with "collatz" are there?
Voorbeeldzinnen.info contains 8 example sentences with "collatz", drawn from a database of millions of English sentences.