Palindromic numbers are numbers which read the same from
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 )
The only numbers which are Square and Tetrahedral are 1, 4 and 19600 (Sloane's A003556)
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.
The sum of two consecutive tetrahedral numbers is a square pyramidal number.
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 :
The basenumber 336 is a number of the form n(n+X):
Other interesting entries from Sloane's table are :
My search program exhaustively scanned all possible palindromic candidates up to length 23.
Basenumber reached 145200000.
[ June 1, 2002 ]
Walter Schneider (email) made a quick search up to basenumber 10^10.
No new palindromic solution was found.
Kevin Brown informs me that he has more info about tetrahedral palindromes in other base representations.
Read his article :
On General Palindromic Numbers
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