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# Achilles number

An Achilles number is a number that is powerful but not a perfect power.[1] A positive integer n is a powerful number if, for every prime divisor or factor p of n, p^{2} is also a divisor. In other words, every prime factor appears at least squared. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1.

Achilles numbers are named after Achilles, a hero of the Trojan war, who was also powerful but imperfect.

Sequence of Achilles numbers

A number *n* = *p*_{1}^{a1}*p*_{2}^{a2}…*p*_{k}^{ak} is powerful if min(*a*_{1}, *a*_{2}, …, *a*_{k}) ≥ 2. If in addition gcd(*a*_{1}, *a*_{2}, …, *a*_{k}) = 1 the number is an Achilles number.

The Achilles numbers up to 5000 are:

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 (sequence A052486 in OEIS).

The smallest pair of consecutive Achilles numbers is:[2]

5425069447 = 73 × 412 × 972

5425069448 = 23 × 260412

Examples

108 is a powerful number. Its prime factorization is 22 · 33, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number.

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power:

\( 784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2. \, \)

So it is not an Achilles number.

References

^ Weisstein, Eric W., "Achilles Number" from MathWorld.

^ Carlos Rivera, The Prime Puzzles and Problem Connection, Problem 53

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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