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Palindromic Tetrahedrals
Factorization Palindromic Cubes

Palindromic numbers are numbers which read the same from

  left to right (forwards) as from the right to left (backwards)

Here are a few random examples : 7, 3113, 44611644

Tetrahedral numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base x ( base + 1 ) x ( base + 2 ) ------------------------------------------ 6

The only numbers which are Square and Tetrahedral are 1, 4 and 19600 (Sloane's A003556)

Tetrahedral Number as proved by Meyl (1878; cited in Dickson 1952, p.25).

On sci.math there was a discussion a while ago about a nonpalindromic relationship :

Question : What numbers are both Triangular and Tetrahedral ?
Answer : The only possible ones are 0, 1, 10, 120, 1540 and 7140 (Sloane's A027568).

Triangular basenumbers are 0, 1, 4, 15, 55 and 119

Tetrahedral basenumbers are 0, 1, 3, 8, 20 and 34

Proof : E.T. Avanesov (Rumanian?) proved in 1966 that the above numbers constitute the complete set of solutions.
Acta Arithmetica, vol. 12, 409-420. The proof is particularly abstruse.

All tetrahedral numbers can only end with 1, 4, 5, 6 or 9. Confert the square numbers !

The product of three consecutive integers is always divisible by 6.

A good source for such statements is “The Book of Numbers”
by John H. Conway and Richard K. Guy.
Click on the image on the left for more background about the book.
The book can be ordered at 'www.amazon.com'.

The sums of the consecutive triangular numbers (starting from 1) are the tetrahedral numbers.

For example : 969 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 + 120 + 136 + 153

or the seventeenth tetrahedral number (17 sums).

The sum of two consecutive tetrahedral numbers is a square pyramidal number.

T_n = 35 and T_n+1 = 56

Then we have 35 + 56 = 91 which is 1² + 2² + 3² + 4² + 5² + 6².

The basenumber 336 leads us not only to a palindromic tetrahedral (still the record number despite its small size!)
     but also to primes in one of the following manners :

336 = 6 x 7 x 8 a product of three consecutives

336 + 1 = 337 is prime.

336² + 337² + 338² = 340709 is prime.

The basenumber 336 is a number of the form n(n+X):

336 = n(n+5) whereby n = 16.

Other interesting entries from Sloane's table are :

%N Tetrahedral numbers which are sum of 2 tetrahedrals. under A002311

%N Not the sum of 4 tetrahedrals (a finite sequence). under A000797

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular tetrahedral numbers are categorised : %N Tetrahedral numbers: C(n+3,3). under A000292. The palindromic tetrahedrals are categorised : %N nth tetrahedral number n(n+1)(n+2)/6 is palindromic under A006029. %N Palindromic tetrahedral numbers under A006030. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

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An alternative way to get a 'structural' insight as how to imagine tetrahedrals is to visit these sites :

Tetrahedrals	From Eric Weisstein's Math Encyclopedia
kite-site	A real life application → kite-site

Index Nr	Info	Basenumber	Length
	Palindromic Tetrahedrals		Length
	Formula (n)(n+1)(n+2)/6
6	Info	336	3
	6.378.736		7
5	Info	21	2
	1.771		4
4	Info	Prime!    17	2
	969		3
3	Info	Prime!    2	1
	4		1
2	Info	1	1
	1		1
1	Info	0	1
	0		1

Kevin Brown informs me that he has more info about tetrahedral palindromes in other base representations.
Read his article :
On General Palindromic Numbers

[ June 10, 2021 ]
Patrick De Geest's search program exhaustively scanned all possible palindromic candidates up to length 41.

[ June 1, 2002 ]
Walter Schneider (email) made a quick search up to basenumber 10^10.
No new palindromic solution was found.

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( © All rights reserved ) - Last modified : December 11, 2022.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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