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-rw-r--r--libgo/go/math/big/nat.go590
1 files changed, 178 insertions, 412 deletions
diff --git a/libgo/go/math/big/nat.go b/libgo/go/math/big/nat.go
index 16a87f5c537..6545bc17ed3 100644
--- a/libgo/go/math/big/nat.go
+++ b/libgo/go/math/big/nat.go
@@ -5,18 +5,22 @@
// Package big implements multi-precision arithmetic (big numbers).
// The following numeric types are supported:
//
-// - Int signed integers
-// - Rat rational numbers
+// Int signed integers
+// Rat rational numbers
+// Float floating-point numbers
//
// Methods are typically of the form:
//
-// func (z *Int) Op(x, y *Int) *Int (similar for *Rat)
+// func (z *T) Unary(x *T) *T // z = op x
+// func (z *T) Binary(x, y *T) *T // z = x op y
+// func (x *T) M() T1 // v = x.M()
//
-// and implement operations z = x Op y with the result as receiver; if it
-// is one of the operands it may be overwritten (and its memory reused).
+// with T one of Int, Rat, or Float. For unary and binary operations, the
+// result is the receiver (usually named z in that case); if it is one of
+// the operands x or y it may be overwritten (and its memory reused).
// To enable chaining of operations, the result is also returned. Methods
-// returning a result other than *Int or *Rat take one of the operands as
-// the receiver.
+// returning a result other than *Int, *Rat, or *Float take an operand as
+// the receiver (usually named x in that case).
//
package big
@@ -24,13 +28,7 @@ package big
// These are the building blocks for the operations on signed integers
// and rationals.
-import (
- "errors"
- "io"
- "math"
- "math/rand"
- "sync"
-)
+import "math/rand"
// An unsigned integer x of the form
//
@@ -68,7 +66,7 @@ func (z nat) norm() nat {
func (z nat) make(n int) nat {
if n <= cap(z) {
- return z[0:n] // reuse z
+ return z[:n] // reuse z
}
// Choosing a good value for e has significant performance impact
// because it increases the chance that a value can be reused.
@@ -78,7 +76,7 @@ func (z nat) make(n int) nat {
func (z nat) setWord(x Word) nat {
if x == 0 {
- return z.make(0)
+ return z[:0]
}
z = z.make(1)
z[0] = x
@@ -122,7 +120,7 @@ func (z nat) add(x, y nat) nat {
return z.add(y, x)
case m == 0:
// n == 0 because m >= n; result is 0
- return z.make(0)
+ return z[:0]
case n == 0:
// result is x
return z.set(x)
@@ -148,7 +146,7 @@ func (z nat) sub(x, y nat) nat {
panic("underflow")
case m == 0:
// n == 0 because m >= n; result is 0
- return z.make(0)
+ return z[:0]
case n == 0:
// result is x
return z.set(x)
@@ -218,6 +216,34 @@ func basicMul(z, x, y nat) {
}
}
+// montgomery computes x*y*2^(-n*_W) mod m,
+// assuming k = -1/m mod 2^_W.
+// z is used for storing the result which is returned;
+// z must not alias x, y or m.
+func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
+ var c1, c2 Word
+ z = z.make(n)
+ z.clear()
+ for i := 0; i < n; i++ {
+ d := y[i]
+ c1 += addMulVVW(z, x, d)
+ t := z[0] * k
+ c2 = addMulVVW(z, m, t)
+
+ copy(z, z[1:])
+ z[n-1] = c1 + c2
+ if z[n-1] < c1 {
+ c1 = 1
+ } else {
+ c1 = 0
+ }
+ }
+ if c1 != 0 {
+ subVV(z, z, m)
+ }
+ return z
+}
+
// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
// Factored out for readability - do not use outside karatsuba.
func karatsubaAdd(z, x nat, n int) {
@@ -337,7 +363,7 @@ func karatsuba(z, x, y nat) {
}
}
-// alias returns true if x and y share the same base array.
+// alias reports whether x and y share the same base array.
func alias(x, y nat) bool {
return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
}
@@ -384,7 +410,7 @@ func (z nat) mul(x, y nat) nat {
case m < n:
return z.mul(y, x)
case m == 0 || n == 0:
- return z.make(0)
+ return z[:0]
case n == 1:
return z.mulAddWW(x, y[0], 0)
}
@@ -488,7 +514,7 @@ func (z nat) divW(x nat, y Word) (q nat, r Word) {
q = z.set(x) // result is x
return
case m == 0:
- q = z.make(0) // result is 0
+ q = z[:0] // result is 0
return
}
// m > 0
@@ -504,7 +530,7 @@ func (z nat) div(z2, u, v nat) (q, r nat) {
}
if u.cmp(v) < 0 {
- q = z.make(0)
+ q = z[:0]
r = z2.set(u)
return
}
@@ -543,10 +569,10 @@ func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
u = nil // u is an alias for uIn or v - cannot reuse
}
u = u.make(len(uIn) + 1)
- u.clear()
+ u.clear() // TODO(gri) no need to clear if we allocated a new u
// D1.
- shift := leadingZeros(v[n-1])
+ shift := nlz(v[n-1])
if shift > 0 {
// do not modify v, it may be used by another goroutine simultaneously
v1 := make(nat, n)
@@ -606,385 +632,6 @@ func (x nat) bitLen() int {
return 0
}
-// MaxBase is the largest number base accepted for string conversions.
-const MaxBase = 'z' - 'a' + 10 + 1 // = hexValue('z') + 1
-
-func hexValue(ch rune) Word {
- d := int(MaxBase + 1) // illegal base
- switch {
- case '0' <= ch && ch <= '9':
- d = int(ch - '0')
- case 'a' <= ch && ch <= 'z':
- d = int(ch - 'a' + 10)
- case 'A' <= ch && ch <= 'Z':
- d = int(ch - 'A' + 10)
- }
- return Word(d)
-}
-
-// scan sets z to the natural number corresponding to the longest possible prefix
-// read from r representing an unsigned integer in a given conversion base.
-// It returns z, the actual conversion base used, and an error, if any. In the
-// error case, the value of z is undefined. The syntax follows the syntax of
-// unsigned integer literals in Go.
-//
-// The base argument must be 0 or a value from 2 through MaxBase. If the base
-// is 0, the string prefix determines the actual conversion base. A prefix of
-// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
-// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
-//
-func (z nat) scan(r io.RuneScanner, base int) (nat, int, error) {
- // reject illegal bases
- if base < 0 || base == 1 || MaxBase < base {
- return z, 0, errors.New("illegal number base")
- }
-
- // one char look-ahead
- ch, _, err := r.ReadRune()
- if err != nil {
- return z, 0, err
- }
-
- // determine base if necessary
- b := Word(base)
- if base == 0 {
- b = 10
- if ch == '0' {
- switch ch, _, err = r.ReadRune(); err {
- case nil:
- b = 8
- switch ch {
- case 'x', 'X':
- b = 16
- case 'b', 'B':
- b = 2
- }
- if b == 2 || b == 16 {
- if ch, _, err = r.ReadRune(); err != nil {
- return z, 0, err
- }
- }
- case io.EOF:
- return z.make(0), 10, nil
- default:
- return z, 10, err
- }
- }
- }
-
- // convert string
- // - group as many digits d as possible together into a "super-digit" dd with "super-base" bb
- // - only when bb does not fit into a word anymore, do a full number mulAddWW using bb and dd
- z = z.make(0)
- bb := Word(1)
- dd := Word(0)
- for max := _M / b; ; {
- d := hexValue(ch)
- if d >= b {
- r.UnreadRune() // ch does not belong to number anymore
- break
- }
-
- if bb <= max {
- bb *= b
- dd = dd*b + d
- } else {
- // bb * b would overflow
- z = z.mulAddWW(z, bb, dd)
- bb = b
- dd = d
- }
-
- if ch, _, err = r.ReadRune(); err != nil {
- if err != io.EOF {
- return z, int(b), err
- }
- break
- }
- }
-
- switch {
- case bb > 1:
- // there was at least one mantissa digit
- z = z.mulAddWW(z, bb, dd)
- case base == 0 && b == 8:
- // there was only the octal prefix 0 (possibly followed by digits > 7);
- // return base 10, not 8
- return z, 10, nil
- case base != 0 || b != 8:
- // there was neither a mantissa digit nor the octal prefix 0
- return z, int(b), errors.New("syntax error scanning number")
- }
-
- return z.norm(), int(b), nil
-}
-
-// Character sets for string conversion.
-const (
- lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz"
- uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
-)
-
-// decimalString returns a decimal representation of x.
-// It calls x.string with the charset "0123456789".
-func (x nat) decimalString() string {
- return x.string(lowercaseDigits[0:10])
-}
-
-// string converts x to a string using digits from a charset; a digit with
-// value d is represented by charset[d]. The conversion base is determined
-// by len(charset), which must be >= 2 and <= 256.
-func (x nat) string(charset string) string {
- b := Word(len(charset))
-
- // special cases
- switch {
- case b < 2 || MaxBase > 256:
- panic("illegal base")
- case len(x) == 0:
- return string(charset[0])
- }
-
- // allocate buffer for conversion
- i := int(float64(x.bitLen())/math.Log2(float64(b))) + 1 // off by one at most
- s := make([]byte, i)
-
- // convert power of two and non power of two bases separately
- if b == b&-b {
- // shift is base-b digit size in bits
- shift := trailingZeroBits(b) // shift > 0 because b >= 2
- mask := Word(1)<<shift - 1
- w := x[0]
- nbits := uint(_W) // number of unprocessed bits in w
-
- // convert less-significant words
- for k := 1; k < len(x); k++ {
- // convert full digits
- for nbits >= shift {
- i--
- s[i] = charset[w&mask]
- w >>= shift
- nbits -= shift
- }
-
- // convert any partial leading digit and advance to next word
- if nbits == 0 {
- // no partial digit remaining, just advance
- w = x[k]
- nbits = _W
- } else {
- // partial digit in current (k-1) and next (k) word
- w |= x[k] << nbits
- i--
- s[i] = charset[w&mask]
-
- // advance
- w = x[k] >> (shift - nbits)
- nbits = _W - (shift - nbits)
- }
- }
-
- // convert digits of most-significant word (omit leading zeros)
- for nbits >= 0 && w != 0 {
- i--
- s[i] = charset[w&mask]
- w >>= shift
- nbits -= shift
- }
-
- } else {
- // determine "big base"; i.e., the largest possible value bb
- // that is a power of base b and still fits into a Word
- // (as in 10^19 for 19 decimal digits in a 64bit Word)
- bb := b // big base is b**ndigits
- ndigits := 1 // number of base b digits
- for max := Word(_M / b); bb <= max; bb *= b {
- ndigits++ // maximize ndigits where bb = b**ndigits, bb <= _M
- }
-
- // construct table of successive squares of bb*leafSize to use in subdivisions
- // result (table != nil) <=> (len(x) > leafSize > 0)
- table := divisors(len(x), b, ndigits, bb)
-
- // preserve x, create local copy for use by convertWords
- q := nat(nil).set(x)
-
- // convert q to string s in base b
- q.convertWords(s, charset, b, ndigits, bb, table)
-
- // strip leading zeros
- // (x != 0; thus s must contain at least one non-zero digit
- // and the loop will terminate)
- i = 0
- for zero := charset[0]; s[i] == zero; {
- i++
- }
- }
-
- return string(s[i:])
-}
-
-// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
-// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
-// repeated nat/Word division.
-//
-// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
-// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
-// Recursive conversion divides q by its approximate square root, yielding two parts, each half
-// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
-// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
-// is made better by splitting the subblocks recursively. Best is to split blocks until one more
-// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
-// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
-// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
-// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
-// specific hardware.
-//
-func (q nat) convertWords(s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) {
- // split larger blocks recursively
- if table != nil {
- // len(q) > leafSize > 0
- var r nat
- index := len(table) - 1
- for len(q) > leafSize {
- // find divisor close to sqrt(q) if possible, but in any case < q
- maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
- minLength := maxLength >> 1 // ~= log2 sqrt(q)
- for index > 0 && table[index-1].nbits > minLength {
- index-- // desired
- }
- if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
- index--
- if index < 0 {
- panic("internal inconsistency")
- }
- }
-
- // split q into the two digit number (q'*bbb + r) to form independent subblocks
- q, r = q.div(r, q, table[index].bbb)
-
- // convert subblocks and collect results in s[:h] and s[h:]
- h := len(s) - table[index].ndigits
- r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
- s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
- }
- }
-
- // having split any large blocks now process the remaining (small) block iteratively
- i := len(s)
- var r Word
- if b == 10 {
- // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
- for len(q) > 0 {
- // extract least significant, base bb "digit"
- q, r = q.divW(q, bb)
- for j := 0; j < ndigits && i > 0; j++ {
- i--
- // avoid % computation since r%10 == r - int(r/10)*10;
- // this appears to be faster for BenchmarkString10000Base10
- // and smaller strings (but a bit slower for larger ones)
- t := r / 10
- s[i] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
- r = t
- }
- }
- } else {
- for len(q) > 0 {
- // extract least significant, base bb "digit"
- q, r = q.divW(q, bb)
- for j := 0; j < ndigits && i > 0; j++ {
- i--
- s[i] = charset[r%b]
- r /= b
- }
- }
- }
-
- // prepend high-order zeroes
- zero := charset[0]
- for i > 0 { // while need more leading zeroes
- i--
- s[i] = zero
- }
-}
-
-// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
-// Benchmark and configure leafSize using: go test -bench="Leaf"
-// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
-// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
-var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
-
-type divisor struct {
- bbb nat // divisor
- nbits int // bit length of divisor (discounting leading zeroes) ~= log2(bbb)
- ndigits int // digit length of divisor in terms of output base digits
-}
-
-var cacheBase10 struct {
- sync.Mutex
- table [64]divisor // cached divisors for base 10
-}
-
-// expWW computes x**y
-func (z nat) expWW(x, y Word) nat {
- return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
-}
-
-// construct table of powers of bb*leafSize to use in subdivisions
-func divisors(m int, b Word, ndigits int, bb Word) []divisor {
- // only compute table when recursive conversion is enabled and x is large
- if leafSize == 0 || m <= leafSize {
- return nil
- }
-
- // determine k where (bb**leafSize)**(2**k) >= sqrt(x)
- k := 1
- for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
- k++
- }
-
- // reuse and extend existing table of divisors or create new table as appropriate
- var table []divisor // for b == 10, table overlaps with cacheBase10.table
- if b == 10 {
- cacheBase10.Lock()
- table = cacheBase10.table[0:k] // reuse old table for this conversion
- } else {
- table = make([]divisor, k) // create new table for this conversion
- }
-
- // extend table
- if table[k-1].ndigits == 0 {
- // add new entries as needed
- var larger nat
- for i := 0; i < k; i++ {
- if table[i].ndigits == 0 {
- if i == 0 {
- table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
- table[0].ndigits = ndigits * leafSize
- } else {
- table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
- table[i].ndigits = 2 * table[i-1].ndigits
- }
-
- // optimization: exploit aggregated extra bits in macro blocks
- larger = nat(nil).set(table[i].bbb)
- for mulAddVWW(larger, larger, b, 0) == 0 {
- table[i].bbb = table[i].bbb.set(larger)
- table[i].ndigits++
- }
-
- table[i].nbits = table[i].bbb.bitLen()
- }
- }
- }
-
- if b == 10 {
- cacheBase10.Unlock()
- }
-
- return table
-}
-
const deBruijn32 = 0x077CB531
var deBruijn32Lookup = []byte{
@@ -1041,7 +688,7 @@ func (x nat) trailingZeroBits() uint {
func (z nat) shl(x nat, s uint) nat {
m := len(x)
if m == 0 {
- return z.make(0)
+ return z[:0]
}
// m > 0
@@ -1058,7 +705,7 @@ func (z nat) shr(x nat, s uint) nat {
m := len(x)
n := m - int(s/_W)
if n <= 0 {
- return z.make(0)
+ return z[:0]
}
// n > 0
@@ -1097,12 +744,36 @@ func (z nat) setBit(x nat, i uint, b uint) nat {
panic("set bit is not 0 or 1")
}
-func (z nat) bit(i uint) uint {
- j := int(i / _W)
- if j >= len(z) {
+// bit returns the value of the i'th bit, with lsb == bit 0.
+func (x nat) bit(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
return 0
}
- return uint(z[j] >> (i % _W) & 1)
+ // 0 <= j < len(x)
+ return uint(x[j] >> (i % _W) & 1)
+}
+
+// sticky returns 1 if there's a 1 bit within the
+// i least significant bits, otherwise it returns 0.
+func (x nat) sticky(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
+ if len(x) == 0 {
+ return 0
+ }
+ return 1
+ }
+ // 0 <= j < len(x)
+ for _, x := range x[:j] {
+ if x != 0 {
+ return 1
+ }
+ }
+ if x[j]<<(_W-i%_W) != 0 {
+ return 1
+ }
+ return 0
}
func (z nat) and(x, y nat) nat {
@@ -1176,7 +847,7 @@ func (z nat) xor(x, y nat) nat {
return z.norm()
}
-// greaterThan returns true iff (x1<<_W + x2) > (y1<<_W + y2)
+// greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
func greaterThan(x1, x2, y1, y2 Word) bool {
return x1 > y1 || x1 == y1 && x2 > y2
}
@@ -1245,6 +916,13 @@ func (z nat) expNN(x, y, m nat) nat {
}
// y > 0
+ // x**1 mod m == x mod m
+ if len(y) == 1 && y[0] == 1 && len(m) != 0 {
+ _, z = z.div(z, x, m)
+ return z
+ }
+ // y > 1
+
if len(m) != 0 {
// We likely end up being as long as the modulus.
z = z.make(len(m))
@@ -1255,13 +933,16 @@ func (z nat) expNN(x, y, m nat) nat {
// 4-bit, windowed exponentiation. This involves precomputing 14 values
// (x^2...x^15) but then reduces the number of multiply-reduces by a
// third. Even for a 32-bit exponent, this reduces the number of
- // operations.
+ // operations. Uses Montgomery method for odd moduli.
if len(x) > 1 && len(y) > 1 && len(m) > 0 {
+ if m[0]&1 == 1 {
+ return z.expNNMontgomery(x, y, m)
+ }
return z.expNNWindowed(x, y, m)
}
v := y[len(y)-1] // v > 0 because y is normalized and y > 0
- shift := leadingZeros(v) + 1
+ shift := nlz(v) + 1
v <<= shift
var q nat
@@ -1379,6 +1060,87 @@ func (z nat) expNNWindowed(x, y, m nat) nat {
return z.norm()
}
+// expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
+// Uses Montgomery representation.
+func (z nat) expNNMontgomery(x, y, m nat) nat {
+ var zz, one, rr, RR nat
+
+ numWords := len(m)
+
+ // We want the lengths of x and m to be equal.
+ if len(x) > numWords {
+ _, rr = rr.div(rr, x, m)
+ } else if len(x) < numWords {
+ rr = rr.make(numWords)
+ rr.clear()
+ for i := range x {
+ rr[i] = x[i]
+ }
+ } else {
+ rr = x
+ }
+ x = rr
+
+ // Ideally the precomputations would be performed outside, and reused
+ // k0 = -mˆ-1 mod 2ˆ_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
+ // Iteration for Multiplicative Inverses Modulo Prime Powers".
+ k0 := 2 - m[0]
+ t := m[0] - 1
+ for i := 1; i < _W; i <<= 1 {
+ t *= t
+ k0 *= (t + 1)
+ }
+ k0 = -k0
+
+ // RR = 2ˆ(2*_W*len(m)) mod m
+ RR = RR.setWord(1)
+ zz = zz.shl(RR, uint(2*numWords*_W))
+ _, RR = RR.div(RR, zz, m)
+ if len(RR) < numWords {
+ zz = zz.make(numWords)
+ copy(zz, RR)
+ RR = zz
+ }
+ // one = 1, with equal length to that of m
+ one = one.make(numWords)
+ one.clear()
+ one[0] = 1
+
+ const n = 4
+ // powers[i] contains x^i
+ var powers [1 << n]nat
+ powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
+ powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
+ for i := 2; i < 1<<n; i++ {
+ powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
+ }
+
+ // initialize z = 1 (Montgomery 1)
+ z = z.make(numWords)
+ copy(z, powers[0])
+
+ zz = zz.make(numWords)
+
+ // same windowed exponent, but with Montgomery multiplications
+ for i := len(y) - 1; i >= 0; i-- {
+ yi := y[i]
+ for j := 0; j < _W; j += n {
+ if i != len(y)-1 || j != 0 {
+ zz = zz.montgomery(z, z, m, k0, numWords)
+ z = z.montgomery(zz, zz, m, k0, numWords)
+ zz = zz.montgomery(z, z, m, k0, numWords)
+ z = z.montgomery(zz, zz, m, k0, numWords)
+ }
+ zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
+ z, zz = zz, z
+ yi <<= n
+ }
+ }
+ // convert to regular number
+ zz = zz.montgomery(z, one, m, k0, numWords)
+ return zz.norm()
+}
+
// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
// If it returns true, n is prime with probability 1 - 1/4^reps.
// If it returns false, n is not prime.
@@ -1404,6 +1166,10 @@ func (n nat) probablyPrime(reps int) bool {
}
}
+ if n[0]&1 == 0 {
+ return false // n is even
+ }
+
const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
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