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<head><title>isPrime -- whether a integer, polynomial, or ideal is prime</title>
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<div><h1>isPrime -- whether a integer, polynomial, or ideal is prime</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>isPrime f</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>an <a href="___Z__Z.html">integer</a></span>, or an element in a <a href="___Polynomial__Ring.html">polynomial ring</a>, or an <span>an <a href="___Ideal.html">ideal</a></span> in a polynomial ring</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <a href="_true.html" title="">true</a> if <tt>f</tt> is either a prime integer or an irreducible polynomial and <a href="_false.html" title="">false</a> otherwise</span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : ZZ/2[t];</pre>
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<tr><td><pre>i2 : isPrime(t^2+t+1)
o2 = true</pre>
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<tr><td><pre>i3 : isPrime(t^2+1)
o3 = false</pre>
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<tr><td><pre>i4 : isPrime 101
o4 = true</pre>
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<tr><td><pre>i5 : isPrime 158174196546819165468118574681196546811856748118567481185669501856749
o5 = true</pre>
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<tr><td><pre>i6 : isPrime 158174196546819165468118574681196546811856748118567481185669501856749^2
o6 = false</pre>
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<p>Since <a href="_factor.html" title="factor a ring element or a ZZ-module">factor</a> returns factors guaranteed only to be pseudoprimes, it may be useful to check their primality, as follows.</p>
<table class="examples"><tr><td><pre>i7 : f = factor 28752093487520394720397634653456
4
o7 = 2 109*1831*3014311519*2987077659845341
o7 : Expression of class Product</pre>
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<tr><td><pre>i8 : peek'_2 f
o8 = Product{Power{2, 4}, Power{109, 1}, Power{1831, 1}, Power{3014311519,
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1}, Power{2987077659845341, 1}}</pre>
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<tr><td><pre>i9 : first \ toList f
o9 = {2, 109, 1831, 3014311519, 2987077659845341}
o9 : List</pre>
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<tr><td><pre>i10 : isPrime \ oo
o10 = {true, true, true, true, true}
o10 : List</pre>
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This function can be used also to determine whether an ideal in a polynomial ring is prime.<table class="examples"><tr><td><pre>i11 : R = QQ[a..d];</pre>
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<tr><td><pre>i12 : I = monomialCurveIdeal(R,{1,5,8})
2 2 3 2 2 3 3 5 3 5 4
o12 = ideal (a*c - b d, b*c - a d , b c - a d, c - a*b*d , b - a c)
o12 : Ideal of R</pre>
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<tr><td><pre>i13 : isPrime I
o13 = true</pre>
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<p>Primality testing for integers is handled by <a href="_pari.html" title="">pari</a>.</p>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_factor.html" title="factor a ring element or a ZZ-module">factor</a> -- factor a ring element or a ZZ-module</span></li>
<li><span><a href="_is__Pseudoprime_lp__Z__Z_rp.html" title="whether an integer is a pseudoprime">isPseudoprime</a> -- whether an integer is a pseudoprime</span></li>
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<div class="waystouse"><h2>Ways to use <tt>isPrime</tt> :</h2>
<ul><li>isPrime(Ideal)</li>
<li>isPrime(ZZ)</li>
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