Problems & Puzzles: Puzzles

 Puzzle 662 547716131821 Tryczak reported that 61 and 547716131821 are primes such that are at the same time the sum of consecutive squares and the difference of consecutive cubes Example: 547716131821 = 523314^2+523315^2 = 427285^3-427284^3 Q. Find more primes like these two.

Contributions came from Giovanni Resta, Jan van Delden, Seiji Tomita, Igor Schein, W. Edwin Clark, Emmanuel Vantieghem & J. K. Andersen

***

Resta wrote:

puzzle 662 can be easily faced by solving
the quadratic diophantine equation x^2+(x+1)^2 = (y+1)^3-y^3

The next two primes are
47513986677009248633982421, for x= 4874114620985, y= 3979697923084
and
4655885807254867892895911581, for x=48248760643434 y=39394948099364

while a larger one is
7700762140941507639613998311244546932058317323483876680440
3013146516834175426213800844965898097403766248827048152222
5409687706931909310787785632321131633553288904292114820298
8227251938537099661164896849141356002484662379868990856762
4533633641743436827107297352685550351119323257187671599435
1318065435265702246006824034347497608167745523529386746271
9866036355105347489188144599452818343919840866262186242893
044553935904752438125521

where x and y can be easily obtained.

***

Jan wrote:

Starting with:

x^2+y^2=(y+1)^2+y^2=2y(y+1)+1 and
v^3-w^3=3(w+1)^3-w^3=3w(w+1)+1

one sees that we must have 3*w(w+1)=2*y(y+1), with p=3*w(w+1)+1 prime.

I searched for solutions to this equation and afterwards checked if p is prime.
Using the first few solutions, I deduced a simple rule to predict where the next solution might be.

I found the following extra solutions:  In order of appearance: w,y,p. [x=y+1;v=w+1].

DIGITS:  13 13 26
3979697923 084
4874114620 985
4751398667 7009248633 982421

DIGITS:  14 14 28
3939494809 9364
4824876064 3434
4655885807 2548678928 95911581

DIGITS:  26 26 52
3487591603 5942680547 211484
4271409930 0104476447 147614
3648988558 0383714493 5191946070 0496576571 9863060812 21

DIGITS:  57 57 114
2545826334 3827842741 0745855539 6920796653 2863838221 7015444
3117987746 4889637038 6380396080 7642560379 1811907621 0024845
1944369517 4510652380 0226254183 7039461850 4009938747 2325491818 0171511005\
6436321012 2606108574 9679544382 3190345459 7741

DIGITS:  61 61 122
2444502419 7608485019 1360200508 2657558141 8818196517 6718854340 4
2993891801 7064286274 0019791215 7138621670 1772833723 2038009690 5
1792677624 0649930705 2586287111 5935747692 9191696429 7832870237 4541009366\
8275531191 5537363651 2778199254 9662034370 3551413518 61

DIGITS:  146 146 291
1031279847 3388587255 2124414320 1407119868 4432350225 7116885615 2583597846\
0405138460 7427993415 4161748568 6334063655 5257404573 7258899335 8830026205\
796684
1263054703 9977681178 4975698021 9468750787 6846876891 8289441732 5301299845\
5392750762 8472770290 6909597761 1130813609 7527841097 1408348293 4565333079\
973414
3190614370 5817792750 0545763908 8004979064 5965767861 3751017649 2368032812\
2349943799 9160694647 9823667128 6505760869 5919774187 5555547424 2028251653\
9542923374 7350887454 3445233755 4529916028 8308251027 2547796908 3626624417\
0313463928 6665218202 0377048164 6889893586 9509086805 4246764120 4570261496\
0805357762 1

DIGITS:  215 215 430
5066478770 9484870723 2883888222 6964415527 0495783227 2640169579 2212224339\
4252875823 6594876689 3519265541 7105845407 0764206130 2069971532 7195867121\
4781821044 2705402241 6086340179 1547775349 3290505769 1436102324 6766184832\
64240
6205143890 7335209512 3872611520 6699021859 8048983491 7861290803 2325993374\
8967818239 7470409832 5059506628 3843545815 1037574221 3482760454 6325395052\
7445437835 5642050522 9801151673 7295517857 9452024270 1333167180 6033330909\
97160
7700762140 9415076396 1399831124 4546932058 3173234838 7668044030 1314651683\
4175426213 8008449658 9809740376 6248827048 1522225409 6877069319 0931078778\
5632321131 6335532889 0429211482 0298822725 1938537099 6611648968 4914135600\
2484662379 8689908567 6245336336 4174343682 7107297352 6855503511 1932325718\
7671599435 1318065435 2657022460 0682403434 7497608167 7455235293 8674627198\
6603635510 5347489188 1445994528 1834391984 0866262186 2428930445 5393590475\
2438125521

I checked all w having 2000 digits or less (i.e. all potential p having 4000 digits or less),
or there are solutions outside of my “heuristic approach”.

B.T.W. v should be equal to w+1 in order for p to be prime. Since v^3-w^3=(v-w)(v^2+vw+w^2).

***

Seiji wrote:

Equation (x+1)^3-x^3=(y+1)^2+y^2 becomes to 3X^2-2Y^2=1 by {x=(X-1)/2, y=(Y-1)/2}.

Integer solutions {X,Y} are given as follows,
{X,Y} = {-1/6sqrt(6)(5+2sqrt(6))^n+1/6sqrt(6)(5-2sqrt(6))^n-1/2(5+2sqrt(6))^n-1/2(5-2sqrt(6))^n,
-1/2(5+2sqrt(6))^n-1/2(5-2*sqrt(6))^n-1/4sqrt(6)(5+2sqrt(6))^n+1/4sqrt(6)(5-2*sqrt(6))^n}, n is integer.

Check whether (x+1)^3-x^3 is a prime number.

Finally, five solutions were found where n < 50.

[n, x, y, (x+1)^3-x^3]

[1, 4, 5, 61]
[6, 427284, 523314, 547716131821]
[13, 3979697923084, 4874114620985, 47513986677009248633982421]
[14, 39394948099364, 48248760643434, 4655885807254867892895911581]
[26, 34875916035942680547211484, 42714099300104476447147614,
3648988558038371449351919460700496576571986306081221]

***

Igor wrote:

61

547716131821

47513986677009248633982421

4655885807254867892895911581

3648988558038371449351919460700496576571986306081221

1944369517451065238002262541837039461850400993874723254918180171511005643632
10122606108574967954438231903454597741

1792677624064993070525862871115935747692919169642978328702374541009366827553
1191553736365127781992549662034370355141351861

3190614370581779275005457639088004979064596576786137510176492368032812234994
3799916069464798236671286505760869591977418755555474242028251653954292337473
5088745434452337554529916028830825102725477969083626624417031346392866652182
020377048164688989358695090868054246764120457026149608053577621

7700762140941507639613998311244546932058317323483876680440301314651683417542
6213800844965898097403766248827048152222540968770693190931078778563232113163
3553288904292114820298822725193853709966116489684914135600248466237986899085
6762453363364174343682710729735268555035111932325718767159943513180654352657
0224600682403434749760816774552352938674627198660363551053474891881445994528
18343919840866262186242893044553935904752438125521

***

Edwin wrote:

61
= 5^2 + 6^2
= 5^3 - 4^3

547716131821
= 523314^2 + 523315^2
= 427285^3 - 427284^3

47513986677009248633982421
= 4874114620985^2 + 4874114620986^2
= 3979697923085^3 - 3979697923084^3
4655885807254867892895911581
= 48248760643434^2 + 48248760643435^2
= 39394948099365^3 - 39394948099364^3

***

Emmanuel wrote:

First, I looked for the solutions of the equation  n^2 +(n+1)^2 = (n+1)^3 - n^3.

It was not difficult to find out that the first  n  appeared as the terms of the sequence

a(i) = ( 0, 5, 54, 539, 5340, 52865, 523314, 5180279, ...  )

which satisfies the recursive relation  a(i+1) = 10 a(i) - a(i-1) + 4.

With a bit of theory of algebraic integers it was possible to prove that there are no other solutions.  So, it remained to determine the  n  for which  n^2 + (n-1)^2  is prime.

With Mathematica, the PC found these ones :

61,

547716131821,

47513986677009248633982421,

4655885807254867892895911581,

3648988558038371449351919460700496576571986306081221,

19443695174510652380022625418370394618504009938747232549181801715110056
4363210122606108574967954438231903454597741,

1792677624064993070525862871115935747692919169642978328702374541009366827
5531191553736365127781992549662034370355141351861,

3190614370581779275005457639088004979064596576786137510176492368032812234
9943799916069464798236671286505760869591977418755555474242028251653954292
3374735088745434452337554529916028830825102725477969083626624417031346392
866652182020377048164688989358695090868054246764120457026149608053577621,

7700762140941507639613998311244546932058317323483876680440301314651683417
5426213800844965898097403766248827048152222540968770693190931078778563232
1131633553288904292114820298822725193853709966116489684914135600248466237
9868990856762453363364174343682710729735268555035111932325718767159943513
1806543526570224600682403434749760816774552352938674627198660363551053474
89188144599452818343919840866262186242893044553935904752438125521.

If there is a next prime, it will have more than  8000  digits.

***

Jens wrote:

We want integer solutions to x^2+(x+1)^2 = (y+1)^3-y^3.
This corresponds to 2*x^2 + 2*x - 3*y^2 - 3*y = 0.
x_0 = 0, y_0 = 0 is a solution.
gives an infinite sequence of integer solutions:
x_(n+1) = 5*x_n + 6*y_n + 5
y_(n+1) = 4*x_n + 5*y_n + 4

PARI/GP says x_n^2+(x_n+1)^2 is a proven prime for
n = 1, 6, 13, 14, 26, 57, 61, 146, 216.
The number of digits in the 9 primes:
2, 12, 26, 28, 52, 114, 122, 291, 430.

PrimeForm/GW says there are no other primes for n < 19000.
The first four solutions are:

n x y prime
- - - -----
1 5 4 61 (given in puzzle)
6 523314 427284 547716131821 (given in puzzle)
13 4874114620985 3979697923084 47513986677009248633982421
14 48248760643434 39394948099364 4655885807254867892895911581

***

Farideh Firoozbakht wrote:

We can easily show that all such primes are of the form 10k+1

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