## A prime which is the reverse concatenation of the first n triangular numbers

The sequence of triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, …

631   is a prime number.

10631   is a prime number.

55453628211510631   is a prime number.

786655453628211510631   is a prime number.

10591786655453628211510631   is a prime number.

What comes next?

Posted in Number Puzzles | Tagged | 1 Comment

## Prime index of a palindromic hexagonal number

A hexagonal number is a polygonal number of the form   $n \, (2 \, n - 1)$

$9561677372927686361$   is a 19-digital prime index of the 39-digital palindromic hexagonal number:

$9561677372927686361 \,(2 \times 9561677372927686361 - 1)$
$= \; 182851348367914603505306419763843158281$

## Prime that is sum of consecutive triangular numbers with prime indices — Part 2

Prime that is sum of consecutive triangular numbers with prime indices — Part 1

If we consider the Index numbers 29 and 43, both prime and sum the consecutive triangular numbers with those prime indices the sum is the prime number 176809, there are many such cases with a start and end prime indices, in this case the number of primes in the range inclusive {29, 31, 37, 41, 43} is also a prime number, there are not so many of those and in the case of {3, 7, 353} where 3 and 7 are the indices and 373 is the sum then we find 373 is a palindromic prime, find the next such case.

## 13331 expressible as a sum of the squares of three consecutive odd triangular numbers

The odd triangular numbers are given by :   1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851, …

Formula :   $(2 \, n - 1) \, (2 \, n - 1 - (-1)^{n})/2$

while the even triangular numbers are :   0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560, …

Formula :   $(2 \, n + 1) \, (2 \, n + 1 -(-1)^{n})/2$

$45^2 \; + \; 55^2 \; + \; 91^2 \; = \; 13331$   is a palindromic prime number

$13331$   is a sum of the squares of three consecutive odd triangular numbers.

Posted in Prime Numbers | Tagged | 1 Comment

## Num3er 1258723

A Pentagonal number is a polygonal number of the form   $n \, (3 \, n - 1) \,/ \,2$

$1258723$   is a prime number.

$1258723$   is also prime index of a 13-digital palindromic pentagonal number

$1258723 \, (3 \times 1258723 - 1)/2 \; = \; 2376574756732$

$2376574756732 \; = \; 2^2 \times 11^2 \times 47 \times 83 \times 1258723$

## Prime that is sum of consecutive triangular numbers with prime indices — Part 1

$T_2 \; = \; 3$

then,

$T_2 + T_3 + T_5 + T_7 + ... + T_{31}$
$3+6+15+28+66+91+153+190+276+435+496 = 1759$   is a prime number.

$1759$   is the smallest prime that is sum of the first consecutive triangular numbers with prime indices.

Posted in Prime Numbers | Tagged | 5 Comments

## Some Triangular number identities — Part 2

Prove the following relations, involving triangular numbers;

$T_{k + n} = T_{k} + T_{2 \, n \, m + n}$,     where     $k = 2 \, n \, m^2 + (2 \, n + 1) \,m$

T$_{n} = T_{n-1} + T_{m} + T_{k}$,     where     $n = (m^2 + k^2 + m + k)/2$

$T_{a+n} + T_{b+n} + T_{c+n} + T_{d+n} \; = \; T_{m - a} + T_{m - b} + T_{m - c} + T_{m - d}$

where     $m = n + (a+b+c+d)/2$

$T_{a} + T_{b} = ( \,T_{n^2 + n - 1} + T_{n^2 - n - 1} \,) \, ( \,T_{m^2 + m - 1} + T_{m^2 - m - 1} \,)$

where     $a = n^2 \, m^2 + n \, m - 1$,     $b = n^2 \, m^2 - n \, m - 1$

$T_{7 \, c + 1} + T_{c - 1} = (T_{7 \, n + 1} + T_{n - 1}) (T_{7 \, m + 1} + T_{m - 1})$

where     $c = 5 \,n \,m + n + m$

$T_{n} - T_{m} = 3^{2 \; \alpha} k$,

where     $n = 3^{2 \; \alpha} k + (3^{2 \; \alpha} - 1)/2$,     $m = 3^{2 \; \alpha} k - (3^{2 \; \alpha} + 1)/2$

$T_{n} + T_{m} = T_{m - 3^{2 \; \alpha} k} + T_{m + 3^{2 \; \alpha} k}$

where     $n = 3^{2 \; \alpha} k^2 + (3^{2 \, \alpha} - 1)/2$,     $m = 3^{2 \; \alpha} k^2 - (3^{2 \; \alpha} + 1)/2$

## Some Triangular number identities — Part 1

Prove the following;

$T_{3 \, n+1} = 9 \, T_{n} + 1$

$T_{7 \, n+3} = 49 \, T_{n} + 6$

$T_{5 \, n+1} = T_{3 \, n} + T_{4 \, n+1}$

$T_{5 \, n+3} = T_{4 \, n+2} + T_{3 \, n+2}$

$T_{13 \, n+10} = T_{5 \, n+4} + T_{12 \, n+9}$

$T_{17 \, n+10} = T_{8 \, n+4} + T_{15 \, n+9}$

$T_{3 \, n - 1} = 2 \, T_{2 \, n - 1} + T_{n}$

$T_{3 \, n} = 2 \, T_{2 \, n} + T_{n - 1}$

$T_{3 \, n + 1} = T_{2 \, n} + T_{2 \, n + 1} + T_{n}$

$3 \, T_{n} + 1 = T_{n - 1} + T_{n} + T_{n + 1}$

$T_{4 \, n^2 + 5 \, n + 2} = T_{4 \, n^2 + 5 \, n} + T_{4 \, n + 2}$

$T_{m + n} = T_{m} + T_{n} + m \, n$

$T_{m + n + 1} = T_{m} + T_{n} + (m + 1) \,(n + 1)$

$T_{m + n - 1} = T_{m} + T_{n} + m \, n - m - n$

$T_{n - m} = T_{m} + T_{n} - m \,(n + 1)$

$T_{n - m - 1} = T_{m} + T_{n} - n \,(m + 1)$

$T_{m \, n} = T_{m} \, T_{n} + T_{n-1} \, T_{m-1}$

$T_{m \, n + 1} = T_{m} \, T_{n} + T_{n-1} \, T_{m-1} + m \, n + 1$

$T_{m \, n - 1} = T_{m} \, T_{n} + T_{n-1} \, T_{m-1} - m \, n$

$T_{n^2} = ( \,T_{n} \,)^2 + ( \,T_{n-1} \,)^2$

$T_{n^2 + m^2} = ( \,T_{n} \,)^2 + ( \,T_{n-1} \,)^2 + ( \,T_{m} \,)^2 + ( \,T_{m-1} \,)^2 - m^2 \, n^2$

$T_{n^2 - m^2} = ( \,T_{n} \,)^2 + ( \,T_{n-1} \,)^2 + ( \,T_{m} \,)^2 + ( \,T_{m-1} \,)^2 - m^2 \,(n^2+1)$

## x^2 + y^2 + z^2 = 3 xyz

Find 3 positive integers   $x, \; y$   and   $z$   which satisfy this equation:

Using this solution, find a different solution by changing one and only one of the numbers.

How many solutions does the equation have?

Extensions:

Are all triple solutions   $\{ \, x, \; y, \; z \, \}$    which contain a number,   $N$,    connected to each other on the tree found below?

Adding some triples to the tree:

$x = 1$

$\{1,1,2 \}$
$\{1,2,5 \}$
$\{1,5,13 \}$
$\{1,13,34 \}$
$\{1,34,89 \}$
$\{1,89,233 \}$
$\{1,233,610 \}$
$\{1,610,1597 \}$
$\{1,1597,4181 \}$
$\{1,4181,10946 \}$
$\{1,10946,28657 \}$
$\{1,28657,75025 \}$
$\{1,75025,196418 \}$
………………………………………..
…………………………………………

$x = 2$

$\{2,5,29\}$
$\{2,29,169\}$,   $\{2,169,985\}$
$\{2,985,5741\}$,   $\{2,5741,33461\}$
$\{2,33461,195025\}$,   $\{2,195025,1136689\}$
$\{2,1136689,6625109\}$,   $\{2,6625109,38613965\}$
$\{2,38613965,225058681\}$,   $\{2,225058681,1311738121\}$
$\{2,1311738121,7645370045\}$,   $\{2,7645370045,44560482149\}$
$\{2,44560482149,259717522849\}$,   $\{2,259717522849,1513744654945\}$
$\{2,1513744654945,8822750406821\}$,   $\{2,8822750406821,51422757785981\}$

………………………………………..
…………………………………………

$x = 5$

$\{5,29,433 \}$
$\{5,433,6466 \}$
$\{5,6466,96557 \}$

$\{5,96557,1441889 \}$
$\{5,1441889,21531778 \}$
$\{5,21531778,321534781 \}$

$\{5,321534781,4801489937 \}$
$\{5,4801489937,71700814274 \}$
$\{5,71700814274,1070710724173 \}$

$\{5,1070710724173,15988960048321 \}$
$\{5,15988960048321,238763690000642 \}$
$\{5,238763690000642,3565466389961309 \}$

$\{5,3565466389961309,53243232159418993 \}$
$\{5,53243232159418993,795083016001323586 \}$
$\{5,795083016001323586,11873002007860434797 \}$

$\{5,11873002007860434797,177299947101905198369 \}$
$\{5,177299947101905198369,2647626204520717540738 \}$
$\{5,2647626204520717540738,39537093120708857912701 \}$

$\{5,39537093120708857912701,590408770606112151149777 \}$
$\{5,590408770606112151149777,8816594465970973409333954 \}$
$\{5,8816594465970973409333954,131658508218958488988859533 \}$

$\{5,131658508218958488988859533,1966061028818406361423559041 \}$
$\{5,1966061028818406361423559041,29359256924057136932364526082 \}$
$\{5,29359256924057136932364526082,438422792832038647624044332189 \}$

$\{5,438422792832038647624044332189,6546982635556522577428300456753 \}$
$\{5,6546982635556522577428300456753,97766316740515800013800462519106 \}$
$\{5,97766316740515800013800462519106,1459947768472180477629578637329837 \}$

………………………………………..
…………………………………………

$x = 13$

$\{13,194,7561 \}$
$\{13,7561,294685 \}$
$\{13,294685,11485154 \}$
$\{13,11485154,447626321 \}$
$\{13,447626321,17445941365 \}$
$\{13,17445941365,679944086914 \}$
$\{13,679944086914,26500373448281 \}$
$\{13,26500373448281,1032834620396045 \}$
$\{13,1032834620396045,40254049821997474 \}$
$\{13,40254049821997474,1568875108437505441 \}$
$\{13,1568875108437505441,61145875179240714725 \}$
$\{13,61145875179240714725, 2383120256881950368834 \}$

………………………………………..
…………………………………………

$\{13,34,1325 \}$
$\{13,1325,51641 \}$
$\{13,51641,2012674 \}$
$\{13,2012674,78442645 \}$
$\{13,78442645,3057250481 \}$
$\{13,3057250481,119154326114 \}$
$\{13,119154326114,4643961467965 \}$
$\{13,4643961467965,180995342924521 \}$
$\{13,180995342924521,7054174412588354 \}$
$\{13,7054174412588354,274931806748021285 \}$
$\{13,274931806748021285,10715286288760241761 \}$
$\{13,10715286288760241761,417621233454901407394 \}$

………………………………………..
…………………………………………

$x = 29$

$\{29,169,14701 \}$
$\{29,14701,1278818 \}$
$\{29,1278818,111242465 \}$
$\{29,111242465,9676815637 \}$
$\{29,9676815637,841771717954 \}$
$\{29,841771717954,73224462646361 \}$
$\{29,73224462646361,6369686478515453 \}$
$\{29,6369686478515453,554089499168198050 \}$
$\{29,554089499168198050,48199416741154714897 \}$

………………………………………..
…………………………………………

$\{29,433,37666 \}$
$\{29,37666,3276509 \}$
$\{29,3276509,285018617 \}$
$\{29,285018617,24793343170 \}$
$\{29,24793343170,2156735837173 \}$
$\{29,2156735837173,187611224490881 \}$
$\{29,187611224490881,16320019794869474 \}$
$\{29,16320019794869474,1419654110929153357 \}$
$\{29,1419654110929153357,123493587631041472585 \}$

………………………………………..
…………………………………………

the next values for   $x$   are   $34, \; 89$

## (x^2 – 1) (y^2 – 1) = (z^2 – 1)^2

$(x^2 - 1) \, (y^2 - 1) \; = \; (z^2 - 1)^2$

An alternate form assuming   $x, y$,   and   $z$   are positive:

$x^2 \, y^2 \; - \; x^2 \; - \; y^2 \; + \; 1 \; = \; (z^2 - 1)^2$     or

$x^2 \, y^2 \; + \; 1 \; = \; x^2 \; + \; y^2 \; + \; (z^2 - 1)^2$

Here are the first few solutions:

$(17^2 - 1) \,(3^2 - 1) = (7^2 - 1)^2$
$(99^2 - 1) \,(17^2 - 1) = (41^2 - 1)^2$
$(577^2 - 1) \,(99^2 - 1) = (239^2 - 1)^2$
$(3363^2 - 1) \,(577^2 - 1) = (1393^2 - 1)^2$
$(19601^2 - 1) \,(3363^2 - 1) = (8119^2 - 1)^2$
$(114243^2 - 1) \,(19601^2 - 1) = (47321^2 - 1)^2$
$(665857^2 - 1) \,(114243^2 - 1) = (275807^2 - 1)^2$
$(3880899^2 - 1) \,(665857^2 - 1) = (1607521^2 - 1)^2$
$(22619537^2 - 1) \,(3880899^2 - 1) = (9369319^2 - 1)^2$
$(131836323^2 - 1) \,(22619537^2 - 1) = (54608393^2 - 1)^2$
$(768398401^2 - 1) \,(131836323^2 - 1) = (318281039^2 - 1)^2$
$(4478554083^2 - 1) \,(768398401^2 - 1) = (1855077841^2 - 1)^2$
$(26102926097^2 - 1) \,(4478554083^2 - 1) = (10812186007^2 - 1)^2$
$(152139002499^2 - 1) \,(26102926097^2 - 1) = (63018038201^2 - 1)^2$
$(886731088897^2 - 1) \,(152139002499^2 - 1) = (367296043199^2 - 1)^2$
$(5168247530883^2 - 1) \,(886731088897^2 - 1) = (2140758220993^2 - 1)^2$
$(30122754096401^2 - 1) \,(5168247530883^2 - 1) = (12477253282759^2 - 1)^2$
$(175568277047523^2 - 1) \,(30122754096401^2 - 1) = (72722761475561^2 - 1)^2$
$(1023286908188737^2 - 1) \,(175568277047523^2 - 1) = (423859315570607^2 - 1)^2$

Posted in Number Puzzles | Tagged | 1 Comment