Interactive · Pairings
Split an even number into every pairing, then knock out the multiples.
Type an even number E. It fans out into every summation pairing that adds to E — 1 + 99, 3 + 97, 5 + 95, and so on. Then it walks the first number of each pairing against every odd before it: if some smaller odd divides it (the first number is a multiple of 3, 5, 7…), that first number is composite and the pairing is marked invalid.
E = 10E = 50E = 100E = 210E = 500E = 1000
prime ✓ — no smaller odd divides it÷d — multiple of a smaller odd (or the unit 1)
E
100
total pairings
25
neither works
7
one is off
12
both work
6
invalid pairing
100 = 9 + 91
checking odds before 9: 9 ÷ 3 = 3 ✓ multiple → STOP
Neither works 7
the unit 1 and composites — no odd prime on either side
| # | First | Second |
|---|---|---|
| 1 | 1 ✗ unit | 99 ÷3 |
| 2 | 9 ÷3 | 91 ÷7 |
| 3 | 15 ÷3 | 85 ÷5 |
| 4 | 25 ÷5 | 75 ÷3 |
| 5 | 35 ÷5 | 65 ÷5 |
| 6 | 45 ÷3 | 55 ÷5 |
| 7 | 49 ÷7 | 51 ÷3 |
One is off 12
exactly one side is prime; the other is a multiple of a smaller odd
| # | First | Second |
|---|---|---|
| 1 | 5 ✓ prime | 95 ÷5 |
| 2 | 7 ✓ prime | 93 ÷3 |
| 3 | 13 ✓ prime | 87 ÷3 |
| 4 | 19 ✓ prime | 81 ÷3 |
| 5 | 21 ÷3 | 79 ✓ prime |
| 6 | 23 ✓ prime | 77 ÷7 |
| 7 | 27 ÷3 | 73 ✓ prime |
| 8 | 31 ✓ prime | 69 ÷3 |
| 9 | 33 ÷3 | 67 ✓ prime |
| 10 | 37 ✓ prime | 63 ÷3 |
| 11 | 39 ÷3 | 61 ✓ prime |
| 12 | 43 ✓ prime | 57 ÷3 |
Both work 6
both sides prime ⇒ a real Goldbach pair for E
| # | First | Second |
|---|---|---|
| 1 | 3 ✓ prime | 97 ✓ prime |
| 2 | 11 ✓ prime | 89 ✓ prime |
| 3 | 17 ✓ prime | 83 ✓ prime |
| 4 | 29 ✓ prime | 71 ✓ prime |
| 5 | 41 ✓ prime | 59 ✓ prime |
| 6 | 47 ✓ prime | 53 ✓ prime |
Each row is one pairing that adds to 100. A number is prime ✓ when no smaller odd divides it; it is ÷d when it is a multiple of the smaller odd d; 1 is the unit and counts as not working. Click any row to walk its first number against every odd before it.
What survives are the pairings whose first number no smaller odd can divide — the primes (plus 1). Pair those survivors up and you are back at Goldbach's question of whether two of them always rebuild E. See the folded-strip write-up for where that gets hard.