吉尔布雷斯猜想

1, 0, 2, 2, 2, 2, 2, 2, 4,...

1, 2, 0, 0, 0, 0, 0, 2,...

1, 2, 0, 0, 0, 0, 2,...

1, 2, 0, 0, 0, 2,...

1, 2, 0, 0, 2,...

吉尔布雷斯猜想猜测除了原本质数数列之外，这些数列的首个数都是1，在1958年由Norman O. Gilbreath提出。

更数学化来说，将 d₀(n) 定义为第 n 个质数， dₖ₊₁(n)=|dₖ(n)-dₖ(n+1)|，其中 k 是非负整数， n 是正整数。证明对于所有正整数 j， dⱼ(1)≡ 1。

1993年，安德鲁·欧德里兹科检查了 10¹³ 以下的质数（346,065,536,839行），都符合此猜想。（相关论文为Iterated absolute values of differences of consecutive primes，可在[1] （页面存档备份，存于互联网档案馆）下载。）

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had published the same observations.

Motivating arithmetic

Consider the prime numbers 2,3,5,7,11,13,17,19,23,29,31,…

Computing the absolute value of the difference between term n and term n+1 in this sequence yields the sequence 1,2,2,4,2,4,2,4,6,2,…

If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the following five sequences in this list are 1,0,2,2,2,2,2,2,4,…

• 1,2,0,0,0,0,0,2,…
• 1,2,0,0,0,0,2,…
• 1,2,0,0,0,2,…
• 1,2,0,0,2,…

What Gilbreath—and Proth before him—noticed is that the first term in each series of differences appears to be 1.

The conjecture

Formally, let (pₙ) denote the sequence of prime numbers. We can define the sequence (dₙ^(k)) recursively by （公式略）

Gilbreath's conjecture states that d₁^(k)=1 for all k≥ 1.

Verification and attempted proofs

Several sources write that, as well as observing the pattern of Gilbreath's conjecture, Proth released what he believed to be a proof of the statement that was later shown to be flawed. However, Zachary Chase disputes this, writing that although Proth called the observation a "theorem", there is no evidence that he published a proof, or even a false proof, of it.

In 1993, Andrew Odlyzko verified that d₁^(k) is equal to 1 for k≤ n=3.4× 10¹¹. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0s and 2s for the next n numbers. This implies that the next n rows begin with a 1.

In 2025, Simon Plouffe announced a computational verification for the primes up to 1014. That same year, Jean-Francois Colonna verified up to 2×1014, and in 2026 up to 1.5×1015. The conjecture remains an open problem.

Generalizations

In 1980, Martin Gardner published a conjecture by Hallard Croft that stated that the property of Gilbreath's conjecture (having a 1 in the first term of each difference sequence) should hold more generally for every sequence that begins with 2, subsequently contains only odd numbers, and has a sufficiently low bound on the gaps between consecutive elements in the sequence. This conjecture has also been repeated by later authors. However, it is false: for every initial subsequence of 2 and odd numbers, and every non-constant growth rate, there is a continuation of the subsequence by odd numbers whose gaps obey the growth rate but whose difference sequences fail to begin with 1 infinitely often.

Odlyzko (1993) is more careful, writing of certain heuristic reasons for believing Gilbreath's conjecture that "the arguments above apply to many other sequences in which the first element is a 1, the others even, and where the gaps between consecutive elements are not too large and are sufficiently random." However, he does not give a formal definition of what "sufficiently random" means. Chase (2024) proves an analogue of the conjecture for sequences that begin with 2 and 3 (like the primes) and subsequently have gaps between successive elements aᵢ-aᵢ₋₁ that are drawn uniformly at random from the even integers in the interval [0,f(i)], for functions f that grow sufficiently slowly (significantly more slowly than the gaps between primes).