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Joined 2026-09-21 Message
Incoming links: Unit fraction
Coresilience Created 2026-10-20 Updated 2026-11-05
For an integer , the coresilience iswhere is Euler's totient function.[ref] It is related to the resilience of a denominator , but is not its complement:For a prime number , but , rather than zero. For the composite number fifteen, and .
As , the value is a unit fraction precisely when divides . A composite prime power with cannot qualify: its value is , already a reduced fraction with numerator greater than one. This excludes an entire family of composite numbers before a search.
Denominator Created 2026-10-14 Updated 2026-11-05
The denominator is the nonzero lower entry in a fraction . It specifies the division being performed; with , the unit fraction is the size of one part. Thus and have the same numerator but different values.
An integer fraction can always be written with positive denominator. Dividing both entries by their greatest common divisor gives a reduced fraction, so and represent the same rational number despite different denominators.
A zero denominator is excluded from ordinary rational numbers. The symbol used at the boundary of the Stern-Brocot tree is a formal boundary symbol, not an additional rational number produced by division by zero.
Egyptian fraction Created 2026-10-25 Updated 2026-11-05
An Egyptian fraction expresses a nonnegative rational number as a finite sum of positive unit fractions. A computational representation may permit repeated denominators, while the classical distinct-denominator convention adds a further requirement. The empty sum represents zero. For example, ; each denominator is a positive integer. A representation that permits repetition also has the simple expansion for nonnegative integers and positive .
Greedy Egyptian fraction expansion 2026-10-25
For a positive proper residual fraction , with , choose , append the unit fraction , and replace the residual by in lowest terms. This is a greedy algorithm: the chosen unit fraction is the largest one not exceeding the residual.
The inequalities imply . Thus the numerator of the reduced residual strictly decreases while positive, proving algorithm termination by a nonnegative-integer loop variant. The loop invariant is that the accumulated sum plus the residual is the original rational number. Since and , the new residual is less than . If it remains positive, its next chosen denominator is therefore strictly greater than ; this expansion also satisfies the distinct-denominator convention. The proof does not bound the denominators by the input's fixed-width range. Arbitrary-precision arithmetic, checked conversion, or a different representable expansion can therefore be necessary in code.
For , is equivalent to . The largest allowed unit fraction therefore has the smallest such integer denominator:The residual is . From we obtainThus the nonnegative numerator strictly decreases at each nonfinal step, and reduction to lowest terms can only decrease it further. This loop variant proves termination of the greedy Egyptian fraction expansion. The case already has the empty expansion and requires no division by .
procedure OutRat(m,n)
Out.Int(m)
Out.String("/")
Out.Int(n)
procedure OutEgypt(e)
if e.i = 0 then
Out.String("0")
else
for j := 0 to e.i-1 do
if j > 0 then Out.String("+")
Out.String("1/")
Out.Int(e.x[j])Ordinary unpadded decimal output and the greedy Egyptian fraction expansion give
5/7 = 1/2+1/5+1/70 for that input. The final newline is supplied by the caller. Stored duplicate denominators, if present in another valid representation, are printed as separate unit fraction occurrences.An Egyptian fraction need not be proper: several positive unit fractions may sum to at least one. Accordingly, accumulate a general nonnegative reduced fraction, rather than calling the proper-input-only The loop invariant is , , , , andWriting , the fraction for the extended prefix has numerator and denominator . These are the
Reduce on every intermediate value. First define an exact helper:def toRatExact(e: Array[Int]): (BigInt, BigInt) = {
var n = BigInt(0)
var d = BigInt(1)
var i = 0
while (i < e.length) {
require(e(i) > 0)
val x = BigInt(e(i))
val g = d.gcd(x)
val nextN = n * (x / g) + d / g
val nextD = d * (x / g)
val h = nextN.gcd(nextD)
n = nextN / h
d = nextD / h
i += 1
}
(n, d)
}
def toRat(e: Array[Int]): (Int, Int) = {
val result = toRatExact(e)
require(result._1.isValidInt && result._2.isValidInt,
"Reduced result does not fit (Int, Int)")
(result._1.toInt, result._2.toInt)
}nextN and nextD in the code. Reducing them preserves the value and restores coprimality. The loop variant is . At exit the helper returns the exact sum in lowest terms, including for the empty array.There are three distinct ways to control integer overflow: cancel the common denominator factor before multiplication, reduce after each addition, and use arbitrary-precision arithmetic when the remaining values still exceed fixed-width bounds. Cancellation reduces intermediate size but cannot guarantee that an exact answer fits
(Int, Int). For example, the sum is in lowest terms, whose denominator exceeds Int.MaxValue. No implementation returning that fixed type can represent every permitted input exactly. The wrapper therefore checks this unavoidable result-type limitation instead of silently truncating; toRatExact supplies the unrestricted mathematical result.[ref]Scala 2.12 BigInt API documentation. Checked representability with isValidInt before the potentially truncating toInt conversion
Print separators before all but the first term, so there is neither a leading nor a trailing plus sign:The loop invariant says that exactly the first unit fractions have been printed in order, separated by
def printRat(num: Int, den: Int): Unit = {
print(num.toString + "/" + den.toString)
}
def printEgypt(e: Array[Int]): Unit = {
if (e.isEmpty) print("0")
var i = 0
while (i < e.length) {
if (i > 0) print(" + ")
print("1/" + e(i).toString)
i += 1
}
} + , with no trailing separator. The loop variant is . The empty array is displayed as zero. Since and , the output is 5/7 = 1/2 + 1/5 + 1/70. The source's earlier example with middle denominator and final denominator contains the arithmetic typo; its final printing example is correct. Indeed , so no positive final term could repair that particular prefix. The printing routine reports the generated terms rather than imposing a particular expansion. Unit fraction Created 2026-10-20 Updated 2026-11-05
A unit fraction is for a positive integer . Its numerator is one, so it is already a reduced fraction; the denominator determines its value uniquely. For example, and are unit fractions, while is not.
Every unit fraction lies in , including . A rational number can have several expressions as a sum of unit fractions: , for example. An Egyptian fraction requires such a finite representation, usually with distinct denominators.