An Achilles number is a number that is powerful but not a

perfect power In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...

. A positive integer is a powerful number if, for every

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

of , is also a

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as , where and are positive integers greater than 1. Achilles numbers were named by

Henry Bottomley Henry may refer to: People *Henry (given name) *Henry (surname) * Henry Lau, Canadian singer and musician who performs under the mononym Henry Royalty * Portuguese royalty ** King-Cardinal Henry, King of Portugal ** Henry, Count of Portugal, ...

after

Achilles In Greek mythology, Achilles ( ) or Achilleus ( grc-gre, Ἀχιλλεύς) was a hero of the Trojan War, the greatest of all the Greek warriors, and the central character of Homer's ''Iliad''. He was the son of the Nereid Thetis and Peleus, k ...

, a hero of the

Trojan war In Greek mythology, the Trojan War was waged against the city of Troy by the Achaeans (Greeks) after Paris of Troy took Helen from her husband Menelaus, king of Sparta. The war is one of the most important events in Greek mythology and has ...

, who was also powerful but imperfect. ''Strong Achilles numbers'' are Achilles numbers whose

Euler totient In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

s are also Achilles numbers.

Sequence of Achilles numbers

A number is powerful if . If in addition 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 . The smallest pair of consecutive Achilles numbers is:Carlos Rivera, ''The Prime Puzzles and Problem Connection''
Problem 53
/ref> : 5425069447 = 7³ × 41² × 97² : 5425069448 = 2³ × 26041²

Examples

108 is a powerful number. Its

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

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 , where and are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by 5² = 25. 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. 500 = 2² × 5³ is a strong Achilles number as its Euler totient of 200 = 2³ × 5² is also an Achilles number.

References

{{DEFAULTSORT:Achilles Number Integer sequences Achilles