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Question Number 216995 by Rasheed.Sindhi last updated on 26/Feb/25

$$\mathrm{Find}\mathrm{all}\mathrm{prime}\mathrm{numbers}\mathrm{p}\mathrm{and}\mathrm{q}$$$$\mathrm{such}\mathrm{that}$$$${\mathrm{p}}^2-{\mathrm{q}}^2=2024$$

Answered by Marzuk last updated on 26/Feb/25

$$$$$$(p+q)(p-q)=2024$$$$2024=2\times 1012$$$$=4\times 506$$$$=8\times 253$$$$=88\times 23$$$$=1\times 2024$$$$=22\times 92$$$$=11\times 184$$$$=44\times 46$$$$p+q+p-q=2p$$$$thatmeanssumoftheeachpairhas$$$$tobedivisibleby2$$$$2+1012=1014$$$$4+506=510$$$$8+253=261$$$$88+23=111$$$$1+2024=2025$$$$22+92=114$$$$11+184=195$$$$44+46=90$$$$sowehaveleft4pairs$$$$2\times 1012$$$$4\times 506$$$$22\times 92$$$$44\times 46$$$$letsseeifthedividendisthesumofeach$$$$pairandthedivisoris2thenthequotient$$$$is\in \mathbb{P}$$$$\frac{\mathrm{\Sigma }(2,1012)}{2}\notin \mathbb{P}$$$$\frac{\mathrm{\Sigma }(4,506)}{2}\notin \mathbb{P}$$$$\frac{\mathrm{\Sigma }(22,92)}{2}\notin \mathbb{P}$$$$\frac{\mathrm{\Sigma }(44,46)}{2}\notin \mathbb{P}$$$$So,thesetofallpossibleprimevaluesof$$$$is\mathrm{\varnothing }$$

Commented by Rasheed.Sindhi last updated on 26/Feb/25

$$\mathcal{T}hankssir!$$

Answered by mehdee7396 last updated on 26/Feb/25

$$2024=1012\times 2=506\times 4=46\times 44=92\times 22$$$$1)p+q=1012\mathrm{\&}p-q=2\Rightarrow p=507\mathrm{\&}q=505\times$$$$2)p+q=506\mathrm{\&}p-q=4\Rightarrow p=255\mathrm{\&}q=251\times$$$$3)p+q=46\mathrm{\&}p-q=44\Rightarrow p=45\mathrm{\&}q=1\times$$$$4)p+q=92\mathrm{\&}p-q=22\Rightarrow p=57\mathrm{\&}q=35\times$$$$thereisnoanswer$$$$$$

Commented by Rasheed.Sindhi last updated on 26/Feb/25

$$\mathbb{T}\mathrm{hanx}\mathrm{sir}!$$

Answered by MathematicalUser2357 last updated on 01/Mar/25

$${\mathrm{can}}^{\prime }\mathrm{t}\mathrm{find}\mathrm{all}\mathrm{prime}\mathrm{numbers}p\mathrm{and}q\mathrm{because}\mathrm{I}\mathrm{tried}\mathrm{to}\mathrm{find}\mathrm{it}!$$

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