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?
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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

 
Paul asks:

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.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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)

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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x^2 + y^2 + z^2 = 3 xyz

 
 

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

                         equa

 

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

How many solutions does the equation have?  

Source:   
http://galileo.org/classroom-examples/math/math-fair-problems/puzzles/xyz-tree/

 

Extensions:

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

xyz tree

 
 
 

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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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(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
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 

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