Difference between revisions of "Perfect square"
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| − | An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36. | + | An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect squares are <math>0, 1, 4, 9, 16, 25, 36</math>. |
| − | The sum of the first <math>n</math> square numbers ( | + | The sum of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math> |
| − | An integer <math>n</math> is a perfect square iff it is a [[quadratic residue]] [[modulo]] all but finitely | + | An integer <math>n</math> is a perfect square [[iff]] it is a [[quadratic residue]] [[modulo]] all but finitely [[prime]]s. |
== Perfect Square Trinomials == | == Perfect Square Trinomials == | ||
| + | A type of perfect square is an equation that is a perfect square trinomial. For example, <math>(x+a)^2=x^2+2xa+a^2</math>. | ||
| − | + | Perfect square trinomials are a type of quadratic equation that have <math>3</math> terms and contain <math>1</math> unique root. | |
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| − | Perfect square trinomials are a type of quadratic equation that have | ||
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| + | For any quadratic equation in the form <math>ax^2+bx+c</math>, it is a perfect square trinomial iff <math>b=2\sqrt{ac}</math>. | ||
==See also == | ==See also == | ||
* [[Perfect cube]] | * [[Perfect cube]] | ||
* [[Perfect power]] | * [[Perfect power]] | ||
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{{stub}} | {{stub}} | ||
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| + | [[Category:Definition]] | ||
Latest revision as of 00:14, 29 November 2023
An integer 
is said to be a perfect square if there is an integer 
so that 
. The first few perfect squares are 
.
The sum of the first square numbers (starting with 
) is 
An integer is a perfect square iff it is a quadratic residue modulo all but finitely primes.
Perfect Square Trinomials
A type of perfect square is an equation that is a perfect square trinomial. For example, 
.
Perfect square trinomials are a type of quadratic equation that have 
terms and contain unique root.
For any quadratic equation in the form 
, it is a perfect square trinomial iff 
.
See also
This article is a stub. Help us out by expanding it.
