A Not So F amous Goldbac h Conjecture

Goldbac h to Euler

Petersbur g, Novemb er 18th, 1752

Sir,

y ou kindly wrote me some time ago that the bo oks

whic h Mr. Sp ener addressed to me were already dis-

patc hed from Berlin in July; how ev er, up to this moment

I do not know to which merc hant in Petersburg they were

addressed or if they w ere even left b ehind at Lüb ec k. As

it is not known to me by what shortcuts you obtained

the diﬀerence of the series

1

3

+

1

7

+

1

11

+

1

19

+ . . . and

1

5

+

1

13

+

1

17

+

1

29

+ . . . ,if you hav e already published the

metho d, please tell me where it is to b e found. I take it to

b e certain that the n umber of all primes of the form a

2

+ 1

is inﬁnitely large, even if I cannot immediately prov e it,

and I do not think your reason for doubting it - that the

n umber of primes of the form a

2

+ 1 is inﬁnitely smaller

than the n um b er of all primes - is at all relev an t, since

no inﬁnite num b er can b e taken so small that it is not

inﬁnitely greater than some other inﬁnite. Mersenne as-

suredly did not say that only ten p erfect num b ers are p os-

sible, as he himself indicates eleven, among which y our

eigh th one, Sir, is ho wev er not comprised; he do es not

state either that the n umber of p erfect num b ers is ﬁnite,

but only that no range of num b ers can b e indicated which

is so large that it could not b e devoid of p erfect num bers.

Y ou will be able to see that for y ourself - in case tome

I I of the Commentarii has still not arriv ed - from the

general preface to Mersenne’s Cogitata Physico Mathe-

matica, §19, as cited by Mr.Winsheim. As regards Le-

unesc hloß’s Parado xa mathematica, they were printed at

Heidelb erg in 1658, in o ctav o; ho wev er, I never p ossessed

this b ook myself, but b orro w ed the copy whic h I read in

1716 from the Old City Library at K önigsb erg, and re-

turned it there b efore I departed for the last time in 1718,

so it is imp ossible that you could hav e seen the b ook at

m y place in P etersburg. On the other hand, I remember

- though not with utter certain ty - that some years ago

y ou wrote me from Berlin you had got the b ook, along

with Bongo’s 1591 list, from the Roy al Library . I am not

a ware of an ything further ab out this Leuneschloß than

that I read somewhere he had b een a professor at Heidel-

b erg; if I am not mistaken, he is also referred to by some

as ’Luneschlos’, so his name should hav e to b e lo ok ed up

in the encyclopedias in both spellings. It seems that he

found his paradox on perfect n umbers in Mersenne and

afterw ards, when writing it down from memory , deviated

from Mersenne’s true meaning. I had already observ ed

in one of my earlier letters that no algebraic formula can

yield only prime num b ers; indeed, taking the formula to

b e, for example, x

3

+ bx

2

+ cx + e , it is obvious that

whenev er x is a multiple of the absolute term e , then

(and consequently inﬁnitely often) the formula will yield

a non-prime num b er; but if e w ere to equal 1, I merely

substitute x + p for x , as then the formula is c hanged into

x

3

+ 3 px

2

+ 3 p

2

x + p

3

+ bx

2

+ 2 bpx + bp

2

+ cx + cp

+ 1

(1)

and whenever x is a multiple of the num ber p

3

+ bp

2

+

cp + 1 , the form ula yields a non-prime n um b er. No w

since this case where the highest p o wer of x equals 3 can

b e extended to all other whims of nature, whatev er the

p o wer of x is, it is impossible to indicate an algebraic

series in whic h there should not occur inﬁnitely many

terms consisting of non-prime n umbers. I hav e yet an-

other small theorem to add, whic h is quite new and which

I take to b e true un til the contrary is pro ved: Any o dd

n umber equals the sum of t wice a square and a prime,

or: 2 n − 1 = 2 a

2

+ p , where a is to signify a whole n um-

b er, including 0, and p a prime n um b er; for example,

17 = 2 · 0

2

+ 17; 21 = 2 · 1

2

+ 19 ; 27 = 2 · 2

2

+ 19 , and

so on. With a dutiful recommendation to y our dearest

family , I remain, Sir, y our most dev oted serv an t

Goldbac h.