Gilbreath's conjecture

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In 1958, N. L. Gilbreath made the following conjecture:

Conjecture (Gilbreath's conjecture, 1958). (Norman L. Gilbreath)

After the first row (the zeroth absolute differences of primes), the successive absolute differences of primes always produce 1 as the leading term.

A054977 Gilbreath transform of the sequence of primes. (a(0)=2, a(n)=1, n >= 1.) (a(n) = 1 + 0^n, n >= 0.)

{2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

Successive absolute differences of primes

$n\,$
0	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
1	1	2	2	4	2	4	2	4	6	2	6	4	2	4	6	6	2	6	4	2	6	4	6	8
2	1	0	2	2	2	2	2	2	4	4	2	2	2	2	0	4	4	2	2	4	2	2	2
3	1	2	0	0	0	0	0	2	0	2	0	0	0	2	4	0	2	0	2	2	0	0
4	1	2	0	0	0	0	2	2	2	2	0	0	2	2	4	2	2	2	0	2	0
5	1	2	0	0	0	2	0	0	0	2	0	2	0	2	2	0	0	2	2	2
6	1	2	0	0	2	2	0	0	2	2	2	2	2	0	2	0	2	0	0
7	1	2	0	2	0	2	0	2	0	0	0	0	2	2	2	2	2	0
8	1	?	?	?	?	?	?	?	?	?	?	?	?	?	?	?	?
9	1	?	?	?	?	?	?	?	?	?	?	?	?	?	?	?
10	1	?	?	?	?	?	?	?	?	?	?	?	?	?	?
11	1	?	?	?	?	?	?	?	?	?	?	?	?	?
12	1	?	?	?	?	?	?	?	?	?	?	?	?	?

Sequences of successive absolute differences of primes

A000040 The prime numbers. (Zeroth differences of primes.) (Zeroth absolute differences of primes.)

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, ...}

A001223 Differences between consecutive primes. DIFF(primes). (First differences of primes.) (First absolute differences of primes.)

{1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, ...}

|A036263| Second absolute differences of primes. |DIFF(DIFF(primes))|. (Note the difference: A036263 Second differences of primes. DIFF(DIFF(primes)).)

{1, 0, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 0, 4, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 10, 10, 2, 4, 8, 8, 4, 0, 2, 2, 0, 4, 8, 8, 2, 2, 10, 0, 8, 2, 2, 2, 4, 8, 4, 0, 0, 4, 4, 2, 2, 8, 4, 10, 2, 2, ...}

A036272 Absolute values of differences of absolute values of second differences of primes. |DIFF(|DIFF(DIFF(primes))|)|. (Third absolute differences of primes.)

{1, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 4, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 8, 0, 8, 2, 4, 0, 4, 4, 2, 0, 2, 4, 4, 0, 6, 0, 8, 10, 8, 6, 0, 0, 2, 4, 4, 4, 0, 4, 0, 2, 0, 6, 4, 6, 8, 0, 8, 2, ...}

A036273 Absolute values of differences of A036272. |DIFF(|DIFF(|DIFF(DIFF(primes))|)|)|. (Fourth absolute differences of primes.)

{1, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 4, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 8, 8, 8, 6, 2, 4, 4, 0, 2, 2, 2, 2, 0, 4, 6, 6, 8, 2, 2, 2, 6, 0, 2, 2, 0, 0, 4, 4, 4, 2, 2, 6, 2, 2, 2, 8, 8, 6, 2, 0, ...}

A036274 Absolute values of differences of A036273. |DIFF(|DIFF(|DIFF(|DIFF(DIFF(primes))|)|)|)|. (Fifth absolute differences of primes.)

{1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0, 8, 0, 0, 2, 4, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, 0, 2, 6, 0, 0, 4, 6, 2, 0, 2, 0, 4, 0, 0, 2, 0, 4, 4, 0, 0, 6, 0, 2, 4, 2, 2, ...}

A?????? Absolute values of differences of A036274. |DIFF(|DIFF(|DIFF(|DIFF(|DIFF(DIFF(primes))|)|)|)|)|. (Sixth absolute differences of primes.)

{1, ...}

Sequences related to Gilbreath's conjecture

A000232 Construct a triangle as in A036262. Sequence is one less than the position of the first number larger than 2 in the n-th row (n-th difference, n >= 1).

{3, 8, 14, 14, 25, 24, 23, 22, 25, 59, 98, 97, 98, 97, 174, 176, 176, 176, 176, 291, 290, 289, 740, 874, 873, 872, 873, 872, 871, 870, 869, 868, 867, 866, 2180, 2179, 2178, ...}

A036277 Position of first term > 2 in n-th row of Gilbreath array shown in A036262, n >= 0. (For n >= 1, A036277(n) = A000232(n) + 1.)

{2, 4, 9, 15, 15, 26, 25, 24, 23, 26, 60, 99, 98, 99, 98, 175, 177, 177, 177, 177, 292, 291, 290, 741, 875, 874, 873, 874, 873, 872, 871, 870, 869, 868, 867, 2181, 2180, 2179, ...}

A001549 Related to Gilbreath conjecture. (Could an OEIS editor please find a more descriptive title? — Daniel Forgues 07:39, 4 June 2012 (UTC))

{4, 10, 17, 18, 30, 34, 69, 109, 111, 189, 192, 193, 194, 195, 311, 763, 898, 900, 2215, 2810, 2811, 2812, 2813, 3417, 4260, 6000, 6002, 6003, 6004, 23331, 31569, 31601, 31602, ...}

Triangle of absolute differences of prime numbers

The successive absolute differences of primes read by anti-diagonals upwards give the infinite sequence of finite sequences (where A036262 is the concatenation thereof)

{{2}, {1, 3}, {1, 2, 5}, {1, 0, 2, 7}, {1, 2, 2, 4, 11}, {1, 2, 0, 2, 2, 13}, {1, 2, 0, 0, 2, 4, 17}, {1, 2, 0, 0, 0, 2, 2, 19}, {1, 2, 0, 0, 0, 0, 2, 4, 23}, {1, 2, 0, 0, 0, 0, 0, 2, 6, 29}, {1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 31}, ...}

A173816 The sum of the numbers in the n-th row of Gilbreath's triangle (A036262).

{2, 4, 8, 10, 20, 20, 26, 26, 32, 40, 50, 54, 56, 52, 62, 66, 88, 84, 94, 104, 96, 108, 104, 120, 128, 136, 128, 134, 136, 154, 216, 188, 204, 190, 212, 200, 206, 212, ...}

A173817 The half of sum of the numbers in the n-th row of Gilbreath's triangle (A036262).

{1, 2, 4, 5, 10, 10, 13, 13, 16, 20, 25, 27, 28, 26, 31, 33, 44, 42, 47, 52, 48, 54, 52, 60, 64, 68, 64, 67, 68, 77, 108, 94, 102, 95, 106, 100, 103, 106, 105, 106, 115, 110, ...}

A036262 Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes read by anti-diagonals upwards.

{2, 1, 3, 1, 2, 5, 1, 0, 2, 7, 1, 2, 2, 4, 11, 1, 2, 0, 2, 2, 13, 1, 2, 0, 0, 2, 4, 17, 1, 2, 0, 0, 0, 2, 2, 19, 1, 2, 0, 0, 0, 0, 2, 4, 23, 1, 2, 0, 0, 0, 0, 0, 2, 6, 29, 1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 31, ...}

The successive absolute differences of primes read by anti-diagonals upwards (omitting the initial row of primes) give the infinite sequence of finite sequences (where A036261 is the concatenation thereof)

{{1}, {1, 2}, {1, 0, 2}, {1, 2, 2, 4}, {1, 2, 0, 2, 2}, {1, 2, 0, 0, 2, 4}, {1, 2, 0, 0, 0, 2, 2}, {1, 2, 0, 0, 0, 0, 2, 4}, {1, 2, 0, 0, 0, 0, 0, 2, 6}, {1, 0, 2, 2, 2, 2, 2, 2, 4, 2}, ...}

A173815 Sum of the numbers in the n-th row of Gilbreath's triangle (A036261).

{1, 3, 3, 9, 7, 9, 7, 9, 11, 19, 17, 15, 9, 15, 13, 29, 23, 27, 33, 23, 29, 21, 31, 31, 35, 25, 27, 27, 41, 89, 57, 67, 51, 63, 49, 49, 49, 43, 39, 51, 39, 63, 47, 65, 45, 75, ...}

A036261 Triangle of numbers arising from Gilbreath's conjecture: successive absolute differences of primes (read by anti-diagonals upwards, omitting the initial row of primes).

{1, 1, 2, 1, 0, 2, 1, 2, 2, 4, 1, 2, 0, 2, 2, 1, 2, 0, 0, 2, 4, 1, 2, 0, 0, 0, 2, 2, 1, 2, 0, 0, 0, 0, 2, 4, 1, 2, 0, 0, 0, 0, 0, 2, 6, 1, 0, 2, 2, 2, 2, 2, 2, 4, 2, 1, 0, 0, 2, 0, 2, 0, 2, 0, 4, 6, 1, 0, 0, ...}

The triangle of absolute differences of prime numbers, where the zeroth anti-diagonal are the primes, is shown below. (Here, each entry is the absolute difference between the entry to the right and the entry to the above right.)

                                                                                                                2	
													1		3	
												1		2		5	
											1		0		2		7	
										1		2		2		4		11	
									1		2		0		2		2		13	
								1		2		0		0		2		4		17	
							1		2		0		0		0		2		2		19	
						1		2		0		0		0		0		2		4		23	
					1		2		0		0		0		0		0		2		6		29	
				1		0		2		2		2		2		2		2		4		2		31	
			1		0		0		2		0		2		0		2		0		4		6		37	
		1		0		0		0		2		2		0		0		2		2		2		4		41	
	1		0		0		0		0		2		0		0		0		2		0		2		2		43

Generalized Gilbreath's conjectures

Gilbreath's conjecture for the sequence of Sophie Germain primes

Conjecture: The diagonal of leading successive absolute differences of the Sophie Germain primes consists, except for the initial 2, only of 1's and 3s.

A080209 Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e. the diagonal of leading successive absolute differences of A005384.

{2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, ...}

A005384 Sophie Germain primes p: 2p+1 is also prime.

{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ...}

To be continued...

See also

• Prime difference function

External links

• Weisstein, Eric W., Gilbreath's Conjecture, from MathWorld—A Wolfram Web Resource.