Prime power

Prime powers (short prime powers ) are natural numbers that are a power of a prime number , e.g. B. . ${\ displaystyle p}$ ${\ displaystyle 625 = 5 \ times 5 \ times 5 \ times = 5 ^ {4}}$

Prime powers occur in finite fields . The number of elements in a finite field is always a prime power.

Examples and values

${\ displaystyle 169 = 13 ^ {2}}$
${\ displaystyle 2401 = 7 ^ {4}}$
${\ displaystyle 1024 = 2 ^ {10}}$

The first prime powers are:

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101 ... (Follow A000961 in OEIS )

If you ignore the simple prime numbers, you get:

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331 ... (sequence A025475 in OEIS )

module

${\ displaystyle p \ geq 5, p ^ {2} \ equiv 1 \ mod {\ {2,3,4,6,8,12,24 \}}}$
${\ displaystyle p \ geq 7, p ^ {4} \ equiv 1 \ mod {\ {2,3,4,5,6,8,10,12,15,16,20,24,30,40,48 , 60.80 \}}}$

...

${\ displaystyle p \ geq 23, p ^ {7920} \ equiv 1 \ mod {\ {2-28,30, ... \}}}$

...

generalization

In arbitrary commutative rings with prime powers are generalized by primary ideals and irreducible ideals . In Dedekindingen ideals are primary or irreducible if and only if they are generated by a power of a prime element. ${\ displaystyle 1}$

Others

In the film Cube (1997) prime powers mark those spaces in a cubic labyrinth structure that contain deadly traps.

Web links

Eric W. Weisstein : Prime Power . In: MathWorld (English).