mathematics

Idoneal number

Idoneal number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Idoneal number rather than just read about it. In short: In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite.

Key takeaways

Idoneal number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Idoneal number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Idoneal number from memory before moving on to harder problems.

Reference excerpt

In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.

Definition A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c. It is sufficient to consider the set {n + k2 | 3 . k2 ≤ n ∧ gcd(n, k) = 1}; if all these numbers are of the form p, p2, 2 · p or 2s for some integer s, where p is a prime, then n is idoneal.

Conjecturally complete listing

The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (sequence A000926 in the OEIS). Results of Peter J. Weinberger from 1973 imply that at most two other idoneal numbers exist, and that the list above is complete if the generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).

See also List of unsolved problems in mathematics

Notes

References Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425–430. D. A. Cox (1989). Primes of the Form x2 + ny2. Wiley-Interscience. p. 61. ISBN 0-471-50654-0. L. Euler, "An illustration of a paradox about the idoneal, or suitable, numbers", 1806 G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55–58 and 64. O-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79–91. [Math. Rev. 85m:11019] G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263. P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73–88. A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 188. P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117–124. Ernst Kani, Idoneal Numbers And Some Generalizations, Ann. Sci. Math. Québec 35, No 2, (2011), 197-227.

External links K. S. Brown, Mathpages, Numeri Idonei M. Waldschmidt, Open Diophantine problems Weisstein, Eric W. "Idoneal Number". MathWorld.

Worked examples

Example 1 — a first encounter with Idoneal number

Start with the simplest possible case. Write down what Idoneal number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Idoneal number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Idoneal number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Idoneal number

In research: Idoneal number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Idoneal number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Idoneal number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Leonhard Euler, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Idoneal number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Idoneal number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Idoneal number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Idoneal number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Idoneal number in simple terms?

Why does Idoneal number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Idoneal number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Idoneal number.