Gilbreath’s Difference Field

Write the primes in increasing order, beginning 2, 3, 5, 7, 11. Under each adjacent pair, write its absolute difference: subtract the smaller value from the larger, so the answer is never negative. These five primes produce the row 1, 2, 2, 4.

Apply the same operation to that new row. It gives 1, 0, 2, followed by 1, 2 and then 1. Each row has one fewer entry, so a finite starting list produces a triangle.

The laboratory prints small triangles as numbers and uses coloured cells for larger ones. Every difference row in these calculations begins with 1; the original prime row begins with 2. Gilbreath's conjecture is the unproved claim that the difference rows keep beginning with 1 however far the starting list of primes is extended.

Two things are going on, and only one of them is mysterious.

The triangle is mostly 0s and 2s. Switch the colouring to magnitude and the field turns into a sea of two values with occasional larger islands. Once a region is all 0s and 2s, differencing keeps it that way: |0−0|, |2−2| and |0−2| are 0, 0 and 2. The structure is self-sustaining, and a left edge sitting on top of such a sea has very little to push it around.

The left edge is stuck on odd numbers. This is the part you can prove, and it is the subject of layer 3. The first entry of every row is odd, so it is 1, 3, 5, … The conjecture is only that it never gets past the first of those.

So the question is not "why is the edge so regular?" - half the regularity is forced. The question is why the one remaining degree of freedom is never used.

Look at the parity of the top row. $3-2=1$ is odd. Every later prime gap is a difference of two odd numbers, so every later gap is even. The first row of the triangle therefore has the shape

Now difference that row. The first entry of the next row is $|\text{even}-\text{odd}|$, which is odd. Every later entry is $|\text{even}-\text{even}|$, which is even. The shape is exactly the shape we started with.

So it repeats, forever. The first entry of every row is odd.

That is a complete proof of half the story, and it costs three lines. Notice what it does not give you: an odd number can be 1, or 3, or 5. Nothing above rules out a 3 appearing on row ten billion.

Let $p_1,p_2,p_3,\dots$ be the primes and define the rows

Proved. For every $n\ge 1$, the value $a_{n,1}$ is odd.

Conjectured (Gilbreath, 1958). For every $n\ge 1$,

Two things are worth noticing about the statement. It is about the infinite triangle - every row, not the first ten billion. And it never mentions primality after the first row: what the proof in layer 3 actually used was the parity of the gaps, not the fact that the numbers were prime. Layer 6 takes that observation seriously.

The parity argument in layer 3 is a proof, and you can check it as carefully as you like. Nothing on this page proves Gilbreath's conjecture, because nobody has.

The history is a caution in itself. François Proth published a proof in 1878 - eighty years before Gilbreath noticed the pattern independently - and the argument was later found to be erroneous.

What is known is computational. The conjecture has been verified for the primes below $10^{13}$. That is an enormous check, and it settles exactly nothing:

Checking $10^{13}$ cases tells you about $10^{13}$ cases.

There is a genuine reduction that makes such checks feasible: if a row begins with 1 and the entries after it are all 0s and 2s, that structure propagates to the row below, so verifying a long enough stretch of 0s and 2s certifies many rows at once. That is a real theorem doing real work - and it still only ever certifies finitely much.

Compare this with the 6174 attractor, where the state space is finite and so checking every case genuinely is a proof. The difference between the two pages is not effort or cleverness. It is that one space runs out and the other does not.

The primes may not be the point. The proof in layer 3 used parity, not primality. So build a sequence with the same shape - 2, 3, then odd numbers with random even gaps of roughly the right size - and it should behave the same way. Select the random model above and it does: the edge holds, row after row, in a sequence with no arithmetic content at all.

But not always. Try seed 12345. The edge breaks at row 9, with a 5. The same construction, a different draw, and the pattern simply fails.

That is the most informative thing on this page. In the random model, counterexamples take seconds to find. For the primes, nobody has found one below $10^{13}$. Either the primes are doing something the model does not capture, or we have not looked nearly far enough - and there is currently no way to tell which.

What recent work does and does not settle. Chase, Hunter and Tao analyse a Cramér-style model of exactly this kind and prove deterministic results about it. Those are theorems about the model. The prime case is untouched, and the conjecture stands.

What this page cannot do. It draws only a few hundred rows while the conjecture concerns all of them, so every 1 visible here is a fact without making the sentence "the edge is always 1" a proved statement.