Sums and Differences of Powerful Numbers
Any odd number is a difference of two consecutive squares: (k + 1)2 = k2 + 2k +12, so (k + 1)2 - k2 = 2k + 1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two: (k + 2)2 - k2 = 4k + 4. However, a singly even number, that is, a number divisible by two but not by four, cannot be expressed as a difference of squares. This motivates the question of determining which singly even numbers can be expressed as differences of powerful numbers. Golomb exhibited some representations of this type:
- 2 = 33 − 52
- 10 = 133 − 37
- 18 = 192 − 73 = 32(33 − 52).
It had been conjectured that 6 cannot be so represented, and Golomb conjectured that there are infinitely many integers which cannot be represented as a difference between two powerful numbers. However, Narkiewicz showed that 6 can be so represented in infinitely many ways such as
- 6 = 5473 − 4632,
and McDaniel showed that every integer has infinitely many such representations (McDaniel, 1982).
Erdős conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by Roger Heath-Brown (1987).
Read more about this topic: Powerful Number
Famous quotes containing the words sums, differences, powerful and/or numbers:
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)
“I don’t know what immutable differences exist between men and women apart from differences in their genitals; perhaps there are some other unchangeable differences; probably there are a number of irrelevant differences. But it is clear that until social expectations for men and women are equal, until we provide equal respect for both men and women, our answers to this question will simply reflect our prejudices.”
—Naomi Weisstein (b. 1939)
“Women of my age in America are at the mercy of two powerful and antagonistic traditions. The first is the ultradomestic fifties with its powerful cult of motherhood; the other is the strident feminism of the seventies with its attempt to clone the male competitive model.... Only in America are these ideologies pushed to extremes.”
—Sylvia Ann Hewitt (20th century)
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)